Performance of stochastic restricted and unrestricted two-parameter estimators in linear mixed models

Abstract

In this article, two parameter estimation using penalized likelihood method in the linear mixed model is proposed. In addition, by considering the stochastic linear restriction for the vector of fixed effects parameters we are introduced the stochastic restricted two parameter estimation. Methods are proposed for estimating variance parameters when unknown. Also, the superiority conditions of the two parameter estimator over the best linear unbiased estimator, and the stochastic restricted two parameter estimator over the stochastic restricted best linear unbiased estimator are obtained under the mean square error matrix sense. Methods are proposed for estimating of the biasing parameters. Finally, a simulation study and a numerical example are given to evaluate the proposed estimators.

Keywords: Linear mixed model, two parameter estimation, stochastic restricted two parameter estimation, matrix mean square error.

1. Introduction

Today many datasets lack the assumption of data independence, which is the main presupposition of many statistical models. For example data collected by cluster or hierarchical sampling, lengthwise studies and frequent measurements or in medical research that simultaneously provides data from one or more body members, the assumption of data independence is unacceptable because the data of a cluster, a group, or an individual are interdependent over time [1]. The default requirement for fitting linear models is the assumption of data independence that does not exist so the use of these models although it leads to unbiased estimates but the variance of estimating coefficients is strongly influenced by the default of data independence. In other words if the data are not independent then the standard error and therefore the confidence interval and the result test result will be for non-trust regression coefficients. Therefore in analyzing these data it is necessary to use methods that can consider this dependence. One of the most important ways to solve this problem is linear mixed models which are generalizations of simple linear models that provide the possibility of random and fixed effects with each other. Linear mixed models are used in many fields of physical, biological, medical and social sciences [2-5].

We consider the linear mixed model (LMM) as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y=X\beta +Zu+\epsilon} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q=\displaystyle\sum _{i=1}^{b}{q}_{i}} ,

(1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\times 1}
vector of observations, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{Z}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ {\mathit{\boldsymbol{Z}}}_{1},\ldots ,{\mathit{\boldsymbol{Z}}}_{b}\right]

 with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{Z}}}_{i}}
is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\times {q}_{i}}
design matrix corresponding to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

-th random effects factor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {q}_{i},\, X}

is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\times p}
observed design matrix for the fixed effects, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta}
 is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p\times 1}
parameter vector of unknown fixed effects, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}}={\left[ {\mathit{\boldsymbol{u}}}_{1}^{'},\ldots ,{\mathit{\boldsymbol{u}}}_{b}^{'}\right] }^{'}}
is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q\times 1}
unobservable vector of random effects and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\epsilon }}}
is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\times 1}
unobservable vector of random errors. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon}
 are independent and have a multivariate normal distribution as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{matrix}u\\\epsilon \end{matrix}\right] \sim \left( \left[ \begin{matrix}0\\0\end{matrix}\right] ,{\sigma }^{2}\left[ \begin{matrix}\mathit{\boldsymbol{G}}(\zeta )&0\\0&W(\xi )\end{matrix}\right] \right)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \zeta}

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi}
 are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {r}_{1}\times 1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {r}_{2}\times 1}
vectors of variance parameters corresponding to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon}

, respectively. Henderson et al [6-7] introduced the set of equations called mixed model equations, and obtained Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{u}}}}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\mathit{\boldsymbol{\beta }}}={\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}\right) }^{-1}{\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \tilde{\mathit{\boldsymbol{u}}}=\mathit{\boldsymbol{G}}{\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}(\mathit{\boldsymbol{y}}- \mathit{\boldsymbol{X}}\tilde{\mathit{\boldsymbol{\beta }}})\right)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}\,(\mathit{\boldsymbol{y}})={\sigma }^{2}\mathit{\boldsymbol{H}}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{H}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{ZG}}{\mathit{\boldsymbol{Z}}}^{'}+\mathit{\boldsymbol{W}} . They Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{u}}}}
are called the best linear unbiased estimator (BLUE) and the best linear unbiased predictor (BLUP), respectively. One of the most common estimators in linear regression is the ordinary least squares (OLS) estimator, which in the case of multicollinearity may lead to estimates with adverse effects such as high variance [8]. To reduce the effects of multicollinearity. Liu et al. [9-10] proposed the ridge estimator and the Liu estimator respectively, which are the well-known alternatives of the OLS estimator. Yang and Chang [11] obtained the two parameter estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\mathit{\boldsymbol{\beta }}}(k,d),}
“Using the mixed estimation technique introduced by Theil et al. [12-13]. They considered the prior information about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\beta }}}
in the form of restriction as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (d-}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k)\overset{\mbox{ˆ}}{\mathit{\boldsymbol{\beta }}}(k)=\mathit{\boldsymbol{\beta }}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{\epsilon }}}_{0},

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k,d}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\mathit{\boldsymbol{\beta }}}(k)}
are respectively the ridge, Liu parameters and the ridge estimator”.

In Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle LMM} , authors such as Gilmour, Jiang and Searl in [14-16], considered a state where the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {X}^{'}{H}^{-1}X}

is singular. Liu and Hu [10] and Eliot et al. [17] inquired the ridge prediction in LMM. Liu and Hu [10] are obtained Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{u}}}(k)}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\mathit{\boldsymbol{\beta }}}(k)={\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+k{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{u}(k)=G{Z}^{'}{H}^{-1}(y-X\tilde{\beta }(k))

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{u}}}(k)}
are the ridge estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\beta }}}
and the ridge predictor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}},}
respectively. Qzkale and Can [18] gave “an example from kidney failure data” to evaluate ridge estimator in linear mixed model. Kuran and Ozkale [19] obtained the mixed and stochastic restricted ridge predictors by using Gilmour approach. They introduced “stochastic linear restriction as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R\beta +\Phi

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\times 1}
vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{R}}}
is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\times p}
known matrix of rank Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\leq p}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Phi}
 is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\times 1}
random vector that is assumed to be distributed with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E(\Phi )=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}\,(\Phi )={\sigma }^{2}\mathit{\boldsymbol{V}}(\mathit{\boldsymbol{v}}),}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\times 1}
vector of variance parameters corresponding to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Phi .}
Also Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Phi}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon}
 are independent”

Then derived the stochastic restricted estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta}

 and the stochastic restricted predictor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}}}
respectively, as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}={\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}\right) }^{-1}\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{r}}\right)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{u}}_{r}=G{Z}^{'}{H}^{-1}\left( y-X{\tilde{\beta }}_{r}\right)

Furthermore, they obtained the stochastic restricted ridge estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta}

 and the stochastic restricted ridge predictor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}}}
respectively, as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\mathit{\boldsymbol{\beta }}},(k)={\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+{\mathit{\boldsymbol{H}}}_{p}\right) }^{-1}\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}}+ {\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{r}}\right)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\mathit{\boldsymbol{u}}}}_{r}(k)=\mathit{\boldsymbol{G}}{\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\left( \mathit{\boldsymbol{y}}- \mathit{\boldsymbol{X}}{\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k)\right)

In this article, we obtain the new two parameter estimations in linear mixed models by taking Yang and Chang’s ideas [11] and considering restriction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (d-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k)\tilde{\beta }(k)=\beta +{\epsilon }_{0}.

In Section Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2,}
we follow the idea of Henderson’s mixed model equations to get the two parameter estimator. Then, by setting stochastic linear restrictions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R\beta +\Phi

 on the vector of fixed effects parameters, we derive the stochastic restricted two parameter estimation. In Section 3, estimates for the variance parameters are obtained when unknown. In Section 4, under the mean square error matrix (MSEM) sense we offer comparisons of new two parameter estimators. In Section 5, Methods are proposed for estimating of the biasing parameters. In Sections 6 and 7, a simulation study and a real data analysis is given. Finally, summary and some conclusions are given in Section 8.

2. The proposed estimators

Under model (1), we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{matrix}u\\y\end{matrix}\right] \sim N\left( \left[ \begin{matrix}0\\X\mathit{\boldsymbol{\beta }}\end{matrix}\right] ,{\sigma }^{2}\left[ \begin{matrix}G&G{Z}^{'}\\ZG&H\end{matrix}\right] \right)

and the joint distribution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h(\mathit{\boldsymbol{y}},\mathit{\boldsymbol{u}})=h(\mathit{\boldsymbol{y}}\mid \mathit{\boldsymbol{u}})h(\mathit{\boldsymbol{u}})= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\mathrm{exp}\,\left\{ \frac{-1}{2{\sigma }^{2}}\left( (\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X\beta }}-\mathit{\boldsymbol{Zu}}{)}^{'}{\mathit{\boldsymbol{W}}}^{-1}(\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X\beta }}-\mathit{\boldsymbol{Zu}})+{\mathit{\boldsymbol{u}}}^{'}{\mathit{\boldsymbol{G}}}^{-1}\mathit{\boldsymbol{u}}\right) \right\} }{{\left. {\left( 2\pi {\sigma }^{2}\right) }^{(n+q)/2}\mathit{\boldsymbol{W}}\right| }^{1/2}\vert \mathit{\boldsymbol{G}}{\vert }^{1/2}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W}
are nonsingular. If the restriction used by Yang and Ghang [11] in linear regression is transferred to linear mixed model, we can produce the two parameter estimator using “penalized term” idea. So by unifying restriction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (d-}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k)\tilde{\beta }(k)=\beta +{\epsilon }_{0}

with model (1) to give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }={X}_{\ast }+{Z}_{\ast }u+{\epsilon }_{\ast }}

(2)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\epsilon }_{0}\sim N\left( 0,{I}_{\ast }\right)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lll} {I}_{\ast }={\sigma }^{2}{I}_{p}, & \quad {y}_{\ast }=\left[ \begin{matrix}y\\(d-k)\tilde{\beta }(k)\end{matrix}\right] ,& \quad {X}_{\ast }=\left[ \begin{matrix}X\\{I}_{p}\end{matrix}\right] ,\\ {Z}_{\star }=\left[ \begin{matrix}Z\\0\end{matrix}\right] ,& \quad {\epsilon }_{\ast }=\left[ \begin{matrix}\epsilon \\{\epsilon }_{0}\end{matrix}\right], &\quad {W}_{\star }=\left[ \begin{matrix}W&0\\0&{I}_{\star }\end{matrix}\right]\end{array}.

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }}
are jointly distributed as 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{matrix}u\\{y}_{\star }\end{matrix}\right] \sim N\left[ \left[ \begin{matrix}0\\{X}_{\star }\end{matrix}\right] \right] ,{\sigma }^{2}\left[ \begin{matrix}G&G{Z}_{\star }^{'}\\{Z}_{\star }G&{H}_{\star }\end{matrix}\right]

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {H}_{\star }=Z,G{Z}_{\star }^{'}+{W}_{\star }=\left[ \begin{matrix}H&0\\0&{I}_{\star }\end{matrix}\right]}

.The conditional distribution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }}
given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }\mid u\sim N\left( X,\beta +\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {Z}_{\ast }u,{W}_{\ast }\right)

 and the logarithm joint density of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}\mathrm{ln}\,f(y,u)&\, =\displaystyle\frac{-1}{2{\sigma }^{2}}\left[ {\left( {y}_{\ast }-{X}_{+}\beta -{Z}_{\ast }u\right) }^{'}{W}_{\ast }^{-1}\left( {y}_{\ast }-{X}_{+}\beta -{Z}_{+}u\right) +{u}^{'}{G}^{-1}u\right] \\ &\,\,\, -\displaystyle\frac{n+p+q}{2}\mathrm{ln}\,\left( 2\pi {\sigma }^{2}\right) -\displaystyle\frac{1}{2}\mathrm{ln}\,\left| {W}_{\ast }\right| -\displaystyle\frac{1}{2}\mathrm{ln}\,\vert G\vert \end{array}

The penalized log-likelihood function is obtained by succession Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast },{X}_{\ast },{Z}_{\ast }}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {W}_{+in}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\left( {y}_{\ast }-{X}_{\ast }\beta -{Z}_{\ast }u\right) }^{'}{W}_{\ast }^{-1}\left( {y}_{\ast }-\right). }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {X}_{\ast }\beta -{Z}_{\ast }u\right) ,

as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}\mathrm{ln}\,f(y,u) & =\displaystyle\frac{-1}{2{\sigma }^{2}}\left[ {y}^{'}{W}^{-1}y+(d-k{)}^{2}{\tilde{\beta }}^{'}(k)\tilde{\beta }(k)-2{\beta }^{'}{X}^{'}{W}^{-1}y-2(d-k){\tilde{\beta }}^{'}(k)\beta \right. \\&\left. -2{u}^{'}{Z}^{-1}y+2{\beta }^{'}{X}^{'}{W}^{-1}Zu+{\beta }^{'}\left( X{W}^{-1}X+{I}_{p}\right) \beta +{u}^{'}{Z}^{'}{W}^{-1}Zu+{u}^{'}{G}^{-1}u\right] \\ &\, -\displaystyle\frac{n+p+q}{2}\mathrm{ln}\,\left( 2\pi {\sigma }^{2}\right) -\displaystyle\frac{1}{2}\mathrm{ln}\,\vert W\vert -\displaystyle\frac{1}{2}\mathrm{ln}\,\vert G\vert \end{array}

(3)

From Eq. (3), we get the partial derivative with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta}

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}},}
then set the equations to zero and by using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }(k,d)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{u}(k,d)}
to denote the solutions give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{I}}}_{p}\right) \tilde{\mathit{\boldsymbol{\beta }}}\left( k,d\right) +{\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}\tilde{\mathit{\boldsymbol{u}}}\left( k,d\right) ={\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{y}}+\left( d-k\right) \tilde{\mathit{\boldsymbol{\beta }}}\left( k\right)

(4)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{Z}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{X}}\tilde{\mathit{\boldsymbol{\beta }}}\left( k,d\right) +\left( {\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}+{\mathit{\boldsymbol{G}}}^{-1}\right) \tilde{\mathit{\boldsymbol{u}}}\left( k,d\right) ={\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{y}}

(5)

By solving the Eq. (5),\tilde{u}(k,d) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}\tilde{u}(k,d) & = {\left( {\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}+{\mathit{\boldsymbol{G}}}^{-1}\right) }^{-1}\left( {\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{y}}-{\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{X}}\tilde{\mathit{\boldsymbol{\beta }}}(k,d)\right) \\ & ={\left( {\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}+{\mathit{\boldsymbol{G}}}^{-1}\right) }^{-1}{\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}(\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X\beta }}(k,d))\end{array}

(6)

Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{u}(k,d)}

into the Eq. (4) we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{matrix}\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{I}}}_{p}\right) \tilde{\mathit{\boldsymbol{\beta }}}(k,d)+\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}{\left( \mathit{\boldsymbol{Z}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}+{\mathit{\boldsymbol{G}}}^{-1}\right) }^{-1}\mathit{\boldsymbol{Z}}{\mathit{\boldsymbol{W}}}^{-1}(\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X}}\tilde{\mathit{\boldsymbol{\beta }}}(k,d))\\=\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{y}}+(d-k)\tilde{\mathit{\boldsymbol{\beta }}}(k)\end{matrix}

(7)

Also using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {H}^{-1}={\left( {Z}^{'}GZ+W\right) }^{-1}={W}^{-1}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {W}^{-1}Z{\left( {Z}^{'}{W}^{-1}Z+{G}^{-1}\right) }^{-1}{Z}^{'}{W}^{-1},

this equation equals to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\beta }(k,d)={\left( {X}^{'}{H}^{-1}X+{I}_{p}\right) }^{-1}\left( {X}^{'}{H}^{-1}y+ (d-k)\tilde{\beta }(k)\right)

(8)

In Eq. (8), if we put Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k)=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+\mathit{\boldsymbol{k}}{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}{\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}},\tilde{\mathit{\boldsymbol{\beta }}}(k,d)

is obtained as follows 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }(k,d)=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. d{\mathit{\boldsymbol{I}}}_{p}\right) {\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+k{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}}

(9)

Due to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( {Z}^{'}{W}^{-1}Z+\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {G}^{-1}\right) G{Z}^{'}={Z}^{'}{W}^{-1}ZG{Z}^{'}+{Z}^{'}={Z}^{'}{W}^{-1}\left( ZG{Z}^{'}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. W\right) ={Z}^{'}{W}^{-1}H,

we get 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\left( {\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}+{\mathit{\boldsymbol{G}}}^{-1}\right) }^{-1}{\mathit{\boldsymbol{G}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{G}}{\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}

(10)

Using Eq. (10), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{u}(k,d)}

equals to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{u}(k,d)=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): GZ{H}^{-1}(y-X\tilde{\beta }(k,d)) .

In section, we obtain the stochastic restricted two parameter estimation. For this, the stochastic Linear restrictions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R\beta +\Phi

 can be unified to model (1) and the restriction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (d-}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k)\tilde{\beta },(k)=\beta +{\epsilon }_{0}

to give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}_{{r}^{\star }}={\mathit{\boldsymbol{X}}}_{{r}^{\star }}\mathit{\boldsymbol{\beta }}+} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{Z}}}_{r}\mathit{\boldsymbol{u}}+{\mathit{\boldsymbol{\epsilon }}}_{\ast ,}

(11)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{{r}^{\ast }}=\left[ \begin{matrix}y\\(d-k){\tilde{\beta }}_{r}(k)\\r\end{matrix}\right] ,{X}_{{r}^{\ast }}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{matrix}X\\{I}_{p}\\R\end{matrix}\right] ,{Z}_{{r}^{\ast }}=\left[ \begin{matrix}Z\\0\\0\end{matrix}\right] ,{\epsilon }_{{r}^{\ast }}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{matrix}\epsilon \\{\epsilon }_{0}\\\Phi \end{matrix}\right]

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {W}_{{r}^{\star }}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{matrix}W&0&0\\0&{I}_{\ast }&0\\0&0&V\end{matrix}\right] .

Then, the conditional distribution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }}
given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}}}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}_{{r}^{\ast }}\mid \mathit{\boldsymbol{u}}\sim \mathit{\boldsymbol{N}}\left( {\mathit{\boldsymbol{X}}}_{{r}^{\ast }}\mathit{\boldsymbol{\beta }}+\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\mathit{\boldsymbol{Z}}}_{{r}^{\ast }}\mathit{\boldsymbol{u}},{\mathit{\boldsymbol{W}}}_{{r}^{\ast }}\right)

 and the logarithm joint density of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{r}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll} \mathrm{ln}\,g(\mathit{\boldsymbol{y}},\mathit{\boldsymbol{u}}) = & \displaystyle\frac{-1}{2{\sigma }^{2}}\left[ {\left( {\mathit{\boldsymbol{y}}}_{{r}^{\ast }}-{\mathit{\boldsymbol{X}}}_{{r}^{\ast }}\mathit{\boldsymbol{\beta }}-{\mathit{\boldsymbol{Z}}}_{{r}^{\ast }}\mathit{\boldsymbol{u}}\right) }^{'}{\mathit{\boldsymbol{W}}}_{{p}^{\ast }}^{-1}\left( {\mathit{\boldsymbol{y}}}_{{r}^{\ast }}-{\mathit{\boldsymbol{X}}}_{{r}^{\ast }}\mathit{\boldsymbol{\beta }}-{\mathit{\boldsymbol{Z}}}_{{r}^{\ast }}\mathit{\boldsymbol{u}}\right) +{\mathit{\boldsymbol{u}}}^{'}{\mathit{\boldsymbol{G}}}^{-1}\mathit{\boldsymbol{u}}\right] \\ & -\displaystyle\frac{n+m+p+q}{2}\mathrm{ln}\,\left( 2\pi {\sigma }^{2}\right) -\displaystyle\frac{1}{2}\mathrm{ln}\,\left| {\mathit{\boldsymbol{W}}}_{r\ast }\right| -\displaystyle\frac{1}{2}\mathrm{ln}\,\vert \mathit{\boldsymbol{G}}\vert \end{array}

Substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{r},{X}_{{r}^{\ast }}{Z}_{{r}^{\ast }}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {W}_{{r}^{+}}}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\left( {y}_{{r}^{\ast }}-{X}_{{r}^{\ast }}\beta -{Z}_{{r}^{\ast }}u\right) }^{'}{W}_{{\gamma }^{\ast }}^{-1}\left( {y}_{{r}^{\ast }}-\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {X}_{{r}^{\ast }}\beta -{Z}_{{r}^{\ast }}u\right) ,

the penalized log-likelihood function is obtained as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{matrix}\mathrm{ln}\,g(y,u)=\displaystyle\frac{-1}{2{\sigma }^{2}}\left[ {y}^{'}{W}^{-1}y+(d-k{)}^{2}{\tilde{\beta }}_{r}^{'}(k){\tilde{\beta }}_{r}(k)+{r}^{'}{V}^{-1}r-2{\beta }^{'}X{W}^{-1}y-2(d-k){\tilde{\beta }}_{r}^{'}(k)\beta \right. \\-2{u}^{'}{Z}^{'}{W}^{-1}y-2{\beta }^{'}{R}^{'}{V}^{-1}r+\beta {R}^{'}{V}^{-1}R\beta +\beta \left( X{W}^{-1}X+{I}_{p}\right) \beta +2{u}^{'}Z{W}^{-1}X\beta \\\left. +{u}^{'}Z{W}^{-1}Zu+{u}^{'}{G}^{-1}u\right] -\displaystyle\frac{n+m+p+q}{2}\mathrm{ln}\,\left( 2\pi {\sigma }^{2}\right) -\displaystyle\frac{1}{2}\mathrm{ln}\,{W}_{{r}^{\star }}\left| -\displaystyle\frac{1}{2}ln\right| G\mid \end{matrix}

(12)

From Eq. (12), we get the partial derivative with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta}

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{u}},}
then set the equations to zero and by using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\beta }}_{r}(k,d)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{u}}_{r}(k,d)}
to denote the solutions give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+{\mathit{\boldsymbol{I}}}_{p}\right) {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)+\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}{\tilde{\mathit{\boldsymbol{u}}}}_{r}(k,d)=\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{y}}+(d-k)\tilde{\mathit{\boldsymbol{\beta }}},(k)+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{r}}

(13)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{X}}{\tilde{\mathit{\boldsymbol{\beta }}}}_{r}\left( k,d\right) +\left( \mathit{\boldsymbol{Z}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{Z}}+{\mathit{\boldsymbol{G}}}^{-1}\right) {\tilde{\mathit{\boldsymbol{u}}}}_{r}\left( k,d\right) =\mathit{\boldsymbol{Z}}{\mathit{\boldsymbol{W}}}^{-1}\mathit{\boldsymbol{y}}

(14)

By solving these equations similar to Eqs. (4) and (5), the following results are obtained

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}{\tilde{\mathit{\boldsymbol{\beta }}}}_{r}\left( k,d\right) & ={\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+d{\mathit{\boldsymbol{I}}}_{p}\right) \\ & \times {\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+k{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{y}}+{\mathit{\boldsymbol{R}}}^{\boldsymbol{'}}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{r}}\right)\end{array}

(15)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\mathit{\boldsymbol{u}}}}_{r}\left( k,d\right) =\mathit{\boldsymbol{G}}{\mathit{\boldsymbol{Z}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\left( \mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X}}{\tilde{\mathit{\boldsymbol{\beta }}}}_{r}\left( k,d\right) \right)

(16)

3. Estimation of variance parameters

In linear mixed models, the variance parameter within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W}
are often unknown that several methods have been proposed by Searle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [16,20-}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 22]

to estimate them. In this section, we estimate the variance parameters using the ML method. The marginal distribution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{\ast }}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N\left( \mathit{\boldsymbol{X}}\cdot \mathit{\boldsymbol{\beta }},{\sigma }^{2}{\mathit{\boldsymbol{H}}}_{\ast }\right) ,}
therefore we can write the marginal log-likelihood function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {l}_{ML}\left( \mathit{\boldsymbol{\beta }},\mathit{\boldsymbol{\phi }},{\mathit{\boldsymbol{y}}}_{\ast }\right) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{-1}{2{\sigma }^{2}}\left\{ {\left( {\mathit{\boldsymbol{y}}}_{\ast }-{\mathit{\boldsymbol{X}}}_{\ast }\mathit{\boldsymbol{\beta }}\right) }^{'}{\mathit{\boldsymbol{H}}}_{\ast }^{-1}\left( {\mathit{\boldsymbol{y}}}_{\ast }-\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left. {\mathit{\boldsymbol{X}}}_{\ast }\mathit{\boldsymbol{\beta }}\right) \right\} - Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{n+p}{2}\mathrm{ln}\,\left( 2\pi {\mathit{\boldsymbol{\sigma }}}^{2}\right) - Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{2}\mathrm{ln}\,\left| {\mathit{\boldsymbol{H}}}_{\ast }\right|

(17)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi ={\left( {\mathit{\boldsymbol{K}}}^{'},{\sigma }^{2}\right) }^{'},\mathit{\boldsymbol{k}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\left( {\zeta }^{'},{\xi }^{'}\right) }^{'}

and are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( {r}_{1}+\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {r}_{2}+1\right) \times 1

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( {r}_{1}+\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {r}_{2}\right) \times 1

vectors of unknown parameters, respectively. Differentiating the Eq. (17) with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta ,{\sigma }^{2}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\kappa }_{j}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,\ldots ,{r}_{1}+{r}_{2}

the partial derivatives is obtained as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {l}_{ML}\left( \mathit{\boldsymbol{\beta }},\mathit{\boldsymbol{\phi }};{\mathit{\boldsymbol{y}}}_{\ast }\right) }{\partial \mathit{\boldsymbol{\beta }}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{-1}{{\sigma }^{2}}\left( {\mathit{\boldsymbol{X}}}_{\ast }^{'}{\mathit{\boldsymbol{H}}}_{\ast }^{-1}{\mathit{\boldsymbol{X}}}_{\ast }\mathit{\boldsymbol{\beta }}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\mathit{\boldsymbol{X}}}_{\ast }^{'}{\mathit{\boldsymbol{H}}}_{\ast }^{-1}{\mathit{\boldsymbol{y}}}_{\ast }\right) ,

(18)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {l}_{ML}\left( \mathit{\boldsymbol{\beta }},\mathit{\boldsymbol{\phi }};{\mathit{\boldsymbol{y}}}_{\ast }\right) }{\partial {\sigma }^{2}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{-(n+p)}{2{\sigma }^{2}}+\frac{{\left( {\mathit{\boldsymbol{y}}}_{\ast }-\mathit{\boldsymbol{X}}\cdot \mathit{\boldsymbol{\beta }}\right) }^{'}{\mathit{\boldsymbol{H}}}_{\cdot }^{-1}\left( {\mathit{\boldsymbol{y}}}_{\ast }-\mathit{\boldsymbol{X}}\cdot \mathit{\boldsymbol{\beta }}\right) }{2{\sigma }^{4}},

(19)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {l}_{ML}\left( \mathit{\boldsymbol{\beta }},\mathit{\boldsymbol{\phi }};{\mathit{\boldsymbol{y}}}_{\ast }\right) }{\partial {\kappa }_{i}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{-1}{2}\mathrm{tr}\,\left( {\mathit{\boldsymbol{H}}}_{\ast }^{-1}{\dot{\mathit{\boldsymbol{H}}}}_{{\ast }_{j}}\right) + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\left( {\mathit{\boldsymbol{y}}}_{\ast }-{\mathit{\boldsymbol{X}}}_{\ast }\mathit{\boldsymbol{\beta }}\right) }^{'}{\mathit{\boldsymbol{H}}}_{\ast }^{-1}{\dot{\mathit{\boldsymbol{H}}}}_{\ast }{\mathit{\boldsymbol{H}}}_{\ast }^{-1}\left( {\mathit{\boldsymbol{y}}}_{\ast }-\mathit{\boldsymbol{X}}\cdot \mathit{\boldsymbol{\beta }}\right) }{2{\sigma }^{2}}

(20)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{H}}_{\ast j}=\partial {H}_{\ast }/\partial {\kappa }_{j}} . Setting Eqs. (18)-(20) equal to zero and using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }(k,d),{\tilde{\sigma }}^{2}(k,d)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{H}}
and instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\beta }},{\sigma }^{2}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{H}}}
gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \&X{\tilde{H}}_{\ast }^{-1}X,\tilde{\beta }(k,d)={X}^{'}{\tilde{H}}_{\ast }^{-1}{y}_{\star }

(21)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (n+p){\tilde{\sigma }}^{2}(k,d)={\left( {y}_{\star }-{X}_{\ast }\tilde{\beta }\left( k,d\right) \right) }^{'}{\tilde{H}}_{\ast }^{-1}\left( {y}_{\star }-{X}_{\star }\tilde{\beta }\left( k,d\right) \right)

(22)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\mathrm{tr}\,\left( {\tilde{H}}_{\ast }^{-1}{\tilde{H}}_{\ast j}\right) =\frac{{\left( {y}_{\star }-{X}_{\ast }\tilde{\beta }\left( k,d\right) \right) }^{'}{\tilde{H}}_{\ast }^{-1}{\tilde{H}}_{\ast j}{\tilde{H}}_{\ast }^{-1}\left( {y}_{\star }-{X}_{\star }\tilde{\beta }\left( k,d\right) \right) }{{\tilde{\sigma }}^{2}\left( k,d\right) }

(23)

Solving Eqs. (21) and (23) yields the estimators

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}\tilde{\mathit{\boldsymbol{\beta }}}(k,d)&={\left( {\mathit{\boldsymbol{X}}}_{\ast }^{'}{\tilde{\mathit{\boldsymbol{H}}}}_{\ast }^{-1}{\mathit{\boldsymbol{X}}}_{\ast }\right) }^{-1}{\mathit{\boldsymbol{X}}}_{\ast }^{'}{\tilde{\mathit{\boldsymbol{H}}}}_{\ast }^{-1}{\mathit{\boldsymbol{y}}}_{\ast }\\ &={\left( {\mathit{\boldsymbol{X}}}^{'}{\tilde{\mathit{\boldsymbol{H}}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( {\mathit{\boldsymbol{X}}}^{'}{\tilde{\mathit{\boldsymbol{H}}}}^{-1}\mathit{\boldsymbol{X}}+d{\mathit{\boldsymbol{I}}}_{p}\right) {\left( {\mathit{\boldsymbol{X}}}^{'}{\tilde{\mathit{\boldsymbol{H}}}}^{-1}\mathit{\boldsymbol{X}}+\mathit{\boldsymbol{k}}{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}{\mathit{\boldsymbol{X}}}^{'}{\tilde{\mathit{\boldsymbol{H}}}}^{-1}\mathit{\boldsymbol{y}}\end{array}

(24)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}{\tilde{\sigma }}^{2}(k,d)&=\displaystyle\frac{1}{(n+p)}{\left( {\mathit{\boldsymbol{y}}}_{\ast }-{\mathit{\boldsymbol{X}}}_{\ast }\tilde{\mathit{\boldsymbol{\beta }}}(k,d)\right) }^{'}{\tilde{\mathit{\boldsymbol{H}}}}_{\cdot }^{-1}\left( {\mathit{\boldsymbol{y}}}_{\ast }-{\mathit{\boldsymbol{X}}}_{\ast }\tilde{\mathit{\boldsymbol{\beta }}}(k,d)\right) \\ & =\displaystyle\frac{1}{(n+p)}\left\{ (\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X}}\tilde{\mathit{\boldsymbol{\beta }}}(k,d){)}^{'}{\tilde{\mathit{\boldsymbol{H}}}}^{-1}(\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{X}}\tilde{\mathit{\boldsymbol{\beta }}}(k,d))\right. \\ & \,\,+ \left. (\tilde{\mathit{\boldsymbol{\beta }}}(k,d)-(d-k)\tilde{\mathit{\boldsymbol{\beta }}}(k){)}^{'}(\tilde{\mathit{\boldsymbol{\beta }}}(k,d)-(d-k)\tilde{\mathit{\boldsymbol{\beta }}}(k))\right\} \end{array}

(25)

Eq. (23) depends on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\sigma }}^{2}(k,d),}
so iterative procedures must be used to solve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_{j}}

's. In the statistical literature, there are four iterative procedures to estimates variance parameters, which include: "Newton-Raphson (NR), Expectation Maximization algorithm (EM), Fisher Scoring (FS) and the Average Information (AI) algorithms". See [22] for details of these procedures. Note that in the stochastic restricted two parameter methods, the ML estimators is obtained similar Eqs. (21),(22) and (23).

4. Comparison of estimators

In this section, we compare the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}
and the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}}
using the mean squares error matrix (MSEM) sense. The estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{2}}
is superior to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{1}}
with respect to the MSEM sense, if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \left( {\tilde{\beta }}_{1},{\tilde{\beta }}_{2}\right) =}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): MSEM\left( {\tilde{\beta }}_{1}\right) -MSEM\left( {\tilde{\beta }}_{2}\right) >0,

that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \left( {\tilde{\beta }}_{1},{\tilde{\beta }}_{2}\right)}
 is a positive definite (pd) matrix. The mean-square error matrix for the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}
is given as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{MSEM}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d),\mathit{\boldsymbol{\beta }})=\mathrm{Var}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))+\mathrm{bias}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))\mathrm{bias}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d){)}^{'}

The variance matrix of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }(k,d)}

is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): V~ar~(\tilde{\beta }(k,d))={\sigma }^{2}{T}_{1}{X}^{'}{H}^{-1}X{T}_{1}^{'}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {T}_{1}={\left( {X}^{'}{H}^{-1}X+{I}_{p}\right) }^{-1}\left( {X}^{'}{H}^{-1}X+\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. d{I}_{p}\right) {\left( {X}^{'}{H}^{-1}X+k{I}_{p}\right) }^{-1}.

The bias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\tilde{\beta }(k,d))}
is given as

bias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\tilde{\beta }(k,d))=E(\tilde{\beta }(k,d)-\beta )=-{\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}{\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+k{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( (k+1-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. d)\mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. k{\mathit{\boldsymbol{I}}}_{p}\right) \mathit{\boldsymbol{\beta }}

The mean-square error matrix for the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\beta }}_{r}(k,d)}

is given as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{MSEM}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d),\mathit{\boldsymbol{\beta }}\right) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\sigma }^{2}{\mathit{\boldsymbol{T}}}_{2}\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}\right) {\mathit{\boldsymbol{T}}}_{2}^{'}+\mathrm{bias}\,\left( {\mathit{\boldsymbol{\beta }}}_{r}(k,d)\right) \mathrm{bias}\,{\left( {\mathit{\boldsymbol{\beta }}}_{r}(k,d)\right) }^{'}

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{T}}}_{2}={\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. d{\mathit{\boldsymbol{I}}}_{p}\right) {\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{\boldsymbol{'}}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+\mathit{\boldsymbol{k}}{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll}{\rm bias}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)\right) & =-{\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}{\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+{\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}+k{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\\ & \times \left( (k+1-d)\left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}+ {\mathit{\boldsymbol{R}}}^{'}{\mathit{\boldsymbol{V}}}^{-1}\mathit{\boldsymbol{R}}\right) + k{\mathit{\boldsymbol{I}}}_{p}\right) \mathit{\boldsymbol{\beta }}\end{array}

4.1. Comparison the Estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}\boldsymbol{(}\mathit{\boldsymbol{k}}\boldsymbol{,}\mathit{\boldsymbol{d}}\boldsymbol{)}} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}} 4.1. Comparison the Estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}\boldsymbol{(}\mathit{\boldsymbol{k}}\boldsymbol{,}\mathit{\boldsymbol{d}}\boldsymbol{)}} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MSEM}

of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{MSEM}\,(\tilde{\mathit{\boldsymbol{\beta }}})=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\sigma }^{2}{\mathit{\boldsymbol{A}}}^{-1},

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{A}}}_{1}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}\right) ,

and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MSEM}
of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}
is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{MSEM}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))={\sigma }^{2}{\mathit{\boldsymbol{T}}}_{1}{\mathit{\boldsymbol{A}}}_{1}{\mathit{\boldsymbol{T}}}_{1}^{'}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\mathit{\boldsymbol{T}}}_{1}{\mathit{\boldsymbol{A}}}_{1}-{\mathit{\boldsymbol{I}}}_{p}\right) \beta {\mathit{\boldsymbol{\beta }}}^{'}{\left( {\mathit{\boldsymbol{T}}}_{1}{\mathit{\boldsymbol{A}}}_{1}-{\mathit{\boldsymbol{I}}}_{p}\right) }^{'}

The estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }(k,d)}

is superior to the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }}
under the MSEM sense, if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{MSEM}\,(\tilde{\mathit{\boldsymbol{\beta }}})-\mathrm{MSEM}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))>0}
then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\Delta }_{1}=MSEM(\tilde{\beta })-MSEM(\tilde{\beta }(k,d))={\sigma }^{2}\left( {A}_{1}^{-1}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {T}_{1}{A}_{1}{T}_{1}\right) -\left( {T}_{1}{A}_{1}-{I}_{p}\right) \beta \beta {\left( {T}_{1}{A}_{1}-{I}_{p}\right) }^{'}>0

According to Farebrother Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1976),}

if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2}\left( {A}_{1}^{-1}-\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {T}_{1}{A}_{1}{T}_{1}\right) >0,

then the necessary and sufficient condition for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}
to be superior to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\beta }}}^{'}{\left( {\mathit{\boldsymbol{T}}}_{1}{\mathit{\boldsymbol{A}}}_{1}-{\mathit{\boldsymbol{I}}}_{p}\right) }^{'}{\left[ {\sigma }^{2}\left( {\mathit{\boldsymbol{A}}}_{1}^{-1}-{\mathit{\boldsymbol{T}}}_{1}{\mathit{\boldsymbol{A}}}_{1}{\mathit{\boldsymbol{T}}}_{1}\right) \right] }^{-1}\left( {\mathit{\boldsymbol{T}}}_{1}{\mathit{\boldsymbol{A}}}_{1}-\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\mathit{\boldsymbol{I}}}_{p}\right) \mathit{\boldsymbol{\beta }}<1 .

5. Selection of parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{k}}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{d}}} 5. Selection of parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{k}}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{d}}}

In the linear regression model, choosing the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

is important so many statisticians were suggested several methods for obtaining this parameter. These methods were proposed by many researchers [23-29] and others. According to Ozkale and Can [18], "we rewrite model (1) in the form of a marginal model in which random effects are not explicitly defined:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}=\mathit{\boldsymbol{X\beta }}+\mathit{\boldsymbol{S}},{\, }\mathit{\boldsymbol{S}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{Zu}}+\mathit{\boldsymbol{\epsilon }}\sim N(0,\mathit{\boldsymbol{H}})

(26)

Because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{H}}}

is pd, then there is a nonsingular symmetric matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{N}}}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{H}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{N}}}^{'}\mathit{\boldsymbol{N}} . If we multiply both sides of model (42) by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{N}}^{-1}} , we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}^{\ast }=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{X}}}^{\ast }\mathit{\boldsymbol{\beta }}+{\mathit{\boldsymbol{S}}}^{\ast } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\mathit{\boldsymbol{y}}}={\mathit{\boldsymbol{N}}}^{-1}\mathit{\boldsymbol{y}},{\mathit{\boldsymbol{X}}}^{\ast }=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{N}}}^{-1}\mathit{\boldsymbol{X}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{S}}}^{\ast }={\mathit{\boldsymbol{N}}}^{-1}\mathit{\boldsymbol{S}}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}\,\left( {\mathit{\boldsymbol{S}}}^{\ast }\right) ={\mathit{\boldsymbol{I}}}_{n}}

. The matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{X}}}^{\ast }{\mathit{\boldsymbol{X}}}^{\ast }=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{X}}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}

is symmetric, so there is an orthogonal matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{P}}}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{P}}}^{'}\mathit{\boldsymbol{P}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{P}}{\mathit{\boldsymbol{P}}}^{'}={\mathit{\boldsymbol{I}}}_{p}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{P}}}^{'}{\mathit{\boldsymbol{X}}}^{\ast }{\mathit{\boldsymbol{X}}}^{\ast }\mathit{\boldsymbol{P}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{\Lambda }=\mathrm{diag}\,\left( {\lambda }_{1},\ldots ,{\lambda }_{p}\right) ,

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\lambda }_{1}\geq {\lambda }_{2}\geq \ldots \geq {\lambda }_{p}}

are the ordered eigenvalues of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{X}}}^{\ast }{\mathit{\boldsymbol{X}}}^{\ast }}

. Then model (26) can be rewritten in a canonical form as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{y}}}^{\ast }=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{X}}}^{\ast \ast }\mathit{\boldsymbol{\gamma }}+{\mathit{\boldsymbol{S}}}^{\ast } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{X}}}^{\ast \ast }=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{X}}}^{\ast }\mathit{\boldsymbol{P}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\gamma }}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{P}}}^{'}\mathit{\boldsymbol{\beta }}'' . Under this model, we get the following representation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\gamma }(k,d)={\left( \boldsymbol{\Lambda }+{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}\left( \boldsymbol{\Lambda }+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. d{\mathit{\boldsymbol{I}}}_{p}\right) {\left( \boldsymbol{\Lambda }+k{\mathit{\boldsymbol{I}}}_{p}\right) }^{-1}{\mathit{\boldsymbol{X}}}^{\ast {\ast }^{'}}{\mathit{\boldsymbol{y}}}^{\ast }

We use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{MSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d),\mathit{\boldsymbol{\beta }})}

to find the optimal values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{MSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d),\mathit{\boldsymbol{\beta }})}

is the mean square error. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}
fixed, the optimal value of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
can be obtained by minimizing the following statement

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{MSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d),\mathit{\boldsymbol{\beta }})= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E\left[ (\tilde{\mathit{\boldsymbol{\beta }}}(k,d)-\mathit{\boldsymbol{\beta }}{)}^{'}(\tilde{\mathit{\boldsymbol{\beta }}}(k,d)-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \mathit{\boldsymbol{\beta }})\right]

Notice that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{MSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d),\mathit{\boldsymbol{\beta }})=\mathrm{MSE}\,(\tilde{\gamma }(k,d),\gamma )=\mathrm{tr}\,[\mathrm{MSEM}\,(\tilde{\gamma }(k,d),\gamma )].}

Therefore, getting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial \mathrm{MSE}\,(\tilde{\gamma }(k,d),\gamma )}{\partial k}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0,

we obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=\frac{{\sigma }^{2}\left( {\lambda }_{i}+d\right) -{\lambda }_{i}{\gamma }_{i}^{2}(1-d)}{{\gamma }_{i}^{2}\left( {\lambda }_{j}+1\right) }.}
Since the optimal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
depends on unknown Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2},}
then according Hoerl and Kennard [9] we can get the estimate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
by substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\gamma }}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\sigma }}^{2}}
as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{k}=\frac{{\tilde{\sigma }}^{2}\left( {\lambda }_{i}+d\right) -{\lambda }_{i}{\tilde{\gamma }}_{i}^{2}(1-d)}{{\tilde{\gamma }}_{i}^{2}\left( {\lambda }_{i}+1\right) }}

(27)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\gamma }}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\sigma }}^{2}}
are the unbiased estimators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\gamma }}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2}}

. According to the estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k,}

which Kibria [25] and Hoerl and Kennard [9] proposed, the harmonic mean value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
in (27) is

Now, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

fixed, and we get the optimal value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}
by minimizing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{MSE}\,(\tilde{\gamma }(k,d),\gamma )}

. Therefore getting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial \mathrm{MSE}\,(\tilde{\gamma }(k,d),\gamma )}{\partial d}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt}=\frac{\sum _{i=1}^{p}\frac{[\left( \left( k+1\right) {\lambda }_{i}+k\right) {\lambda }_{i}{\tilde{\gamma }}_{i}^{2}-\, {\lambda }_{i}^{2}{\tilde{\sigma }}^{2}\, ]}{[{(\, {\lambda }_{i}+k)}^{2}\ast {({\lambda }_{i}\, +1)}^{2}]}}{\sum _{i=1}^{p}\frac{[\left( {\tilde{\sigma }}^{2}\, +\, \, {\lambda }_{i}{\tilde{\gamma }}_{i}^{2}\right) {\lambda }_{i}]}{[{({\lambda }_{i}+k)}^{2}\ast {({\lambda }_{i}+1)}^{2}]}}}

(28)

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

is always be positive, in this section we get the positive condition of the estimator in Eq. (27). For this purpose, we use the following theorem.

Theorem 5.1

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{d}>max\left\{ \frac{1-{\tilde{\sigma }}^{2}{\tilde{\gamma }}_{i}^{2}}{1+{\tilde{\sigma }}^{2}/\left( {\lambda }_{i}{\tilde{\gamma }}_{i}^{2}\right) }\right\}} , for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i,}

then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
are always positive.

Proof.

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{{\sigma }^{2}\left( {\lambda }_{i}+d\right) -{\lambda }_{i}{\gamma }_{i}^{2}(1-d)}{{\gamma }_{i}^{2}\left( {\lambda }_{i}+1\right) }>0} , then the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

are positive. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\gamma }_{i}^{2}\left( {\lambda }_{i}+1\right) >0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2}\left( {\lambda }_{i}+\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. d\right) -{\lambda }_{i}{\gamma }_{i}^{2}(1-d)

must be positive for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

. Then we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d>\frac{1-{\sigma }^{2}/{\gamma }_{i}^{2}}{1+{\sigma }^{2}/\left( {\lambda }_{i}{\gamma }_{i}^{2}\right) }}

and because depends on the unknown parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\gamma }_{i}^{2}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2},}
so their unbiased estimators are replaced. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
is always positive if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{d}}
is selected as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{d}>max\left\{ \frac{1-{\tilde{\sigma }}^{2}{\tilde{\gamma }}_{i}^{2}}{1+{\tilde{\sigma }}^{2}/\left( {\lambda }_{j}{\tilde{\gamma }}_{i}^{2}\right) }\right\}}

.

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{1-{\tilde{\sigma }}^{2}/{\gamma }_{i}^{2}}{1+{\tilde{\sigma }}^{2}/\left( {\lambda }_{i}{\tilde{\gamma }}_{i}^{2}\right) }}

is always less than one and since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}
must be between zero and one, we consider the inequality Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{d}>max\left\{ \frac{1-{\tilde{\sigma }}^{2}{\tilde{\gamma }}_{i}^{2}}{1+{\tilde{\sigma }}^{2}/\left( {\lambda }_{i}{\tilde{\gamma }}_{i}^{2}\right) }\right\}}
as follows

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle max\left\{ \frac{1-{\tilde{\sigma }}^{2}{\tilde{\gamma }}_{i}^{2}}{1+{\tilde{\sigma }}^{2}/\left( {\lambda }_{i}{\tilde{\gamma }}_{i}^{2}\right) },o\right\} <\tilde{d}<1}

(29)

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt~}}

in (28) depends on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
and the estimators of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}

, we consider an iterative method for these parameters by applying the following method. Step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1,}

calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{d}}
from Eq. (29). Step 2, calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
by using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{d}}
in step 1 . Step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3,}
calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt~}}
from Eq. (28) by using the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
in Step 2 . Step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 4,}
if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt~}}
is not between zero and one use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt~}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{d}

6. A simulation study

In this section, we compare the performance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}},\tilde{\mathit{\boldsymbol{\beta }}}(k,d),{\tilde{\mathit{\boldsymbol{\beta }}}}_{t}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)}
with a simulation study. For this purpose, we calculate the estimated mean square error (EMSE) with various values of sample size, variance and degree of collinearity. Following McDonald and Galarneau [27], we are computed the fixed effects as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {X}_{ijc}={\left( 1-{\rho }^{2}\right) }^{1/2}{w}_{ijc}+\rho {w}_{ijp+l},{\, }i=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,\ldots ,t,{\, }j=1,\ldots ,{n}_{i},{\, }c=1,\ldots ,p

(30)

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {w}_{ijc}}

independent standard normal are pseudo-random numbers and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\rho }^{2}}
is the correlation between any two fixed effects. Three different sets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\rho }^{2}}
were considered as 0.75,0.85 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.95.}
The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{Z}}}
matrix is produced in a completely randomized design. Observation on responses are then determined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{ij}={\beta }_{1}{x}_{ij1}+\ldots +{\beta }_{p}{x}_{ijp}+} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {u}_{i1}{z}_{ij1}+\ldots +{u}_{iq}{z}_{ijq}+{\epsilon }_{ij},{\, }i=1,\ldots ,t,j= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,\ldots ,{n}_{i}

(31)

We consider two designs that in the first design Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {n}_{i}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3

and in the second design Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {n}_{i}=7}

. Also, the same value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 9,p=4

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q=9}
are taken in both designs. Following Ozcale and Can (2017) “The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{\beta }}}
vector was chosen as the eigenvector corresponding to the largest eigenvalue of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{X}}^{'}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}}}
matrix”. The variances matrix of random effects Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{i}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{ij}}
are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{G}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\sigma }_{1}^{2}{\mathit{\boldsymbol{I}}}_{q}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{W}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\sigma }^{2}{\mathit{\boldsymbol{I}}}_{a}
respectively. They Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{i}}

are generated from the normal distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N(0,G)}

. We are considered Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.5,1

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }_{1}^{2}=0.5,1}

.The trial was replicated 1000 times by generating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{i}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{ij}.}
For each simulated data set derived Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt~},}
and then the estimated mean squared error (EMSE) calculated as calculated the relative mean square (RMSE) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( \tilde{\mathit{\boldsymbol{\beta }}}:{\tilde{\mathit{\boldsymbol{\beta }}}}^{\ast }\right) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}})}{\mathrm{EMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}^{\ast }\right) }<br/>

when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle RMSE}

is greater than one, it indicates that the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}^{\ast }}
superior to the estimator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

. For the stochastic linear restriction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{R\beta }}+\Phi ,

the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{R}}}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\times p}
and generated from the normal distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N(0,1)}
and the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Phi}
 is generated from the normal distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N(0,\mathit{\boldsymbol{V}})}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{V}}}
is taken as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle =}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{diag}\,\left( {\sigma }^{2}{\mathit{\boldsymbol{I}}}_{m}\right.

 ) with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2 . In tables 1 , we obtain the values of EMSE and RMSE for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}},\tilde{\mathit{\boldsymbol{\beta }}}(k,d),{\tilde{\mathit{\boldsymbol{\beta }}}}_{r}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d).}
We have the following results for Table 1: 

(i) In the whole table, the EMSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}

is less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

. Also, the EMSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)}

is less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}}

. In general, the EMSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)}

is less than all estimators. 

(ii) As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\rho }^{2}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2}}
increase, the EMSE values of the estimators increase.

(iii) As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\rho }^{2}}

increases, difference between the EMSE values of the two parameter estimators and the EMSE values of the best linear unbiased estimators increase. This implies an increase in the improvement of the two-parameter estimators.

Table 1. Estimated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MSE}

and SMSE values with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM},{\tilde{d}}_{opt~}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 9

p=0.75	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\sigma }^{2},{\sigma }_{1}^{2}\right) =(0.5,0.5)		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\sigma }^{2},{\sigma }_{1}^{2}\right) =(1,1)
ni	3	7	3	7
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}})	0.0001526	7.3974e-05	0.0003053	0.0001479
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,\left( {\tilde{{\mathit{\boldsymbol{\beta }}}_{\mathit{\boldsymbol{r}}}}}_{\, }\right)	0.0001461	7.1668e-05	0.0002923	0.0001433
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))	0.0001032	6.9166e-05	0.0001912	0.0001332
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)\right)	9.9335e-05	6.7089e-05	0.0001845	0.0001293
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}:\tilde{\mathit{\boldsymbol{\beta }}}(k,d))	1.4795	1.0695	1.5965	1.1104
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}:{\tilde{\mathit{\boldsymbol{\beta }}}}_{c}(k,d)\right)	1.4716	1.0682	1.5845	1.1083
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( \tilde{\mathit{\boldsymbol{\beta }}}(k,d):{\tilde{\mathit{\boldsymbol{\beta }}}}_{\mathit{\boldsymbol{c}}}(k,d)\right)	1.0389	1.0309	1.0363	1.0301
p=0.85	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\sigma }^{2},{\sigma }_{1}^{2}\right) =(0.5,0.5)		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\sigma }^{2},{\sigma }_{1}^{2}\right) =(1,1)
ni	3	7	3	7
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}})	0.0002198	0.0003460	0.0004396	0.0002387
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,\left( {\tilde{{\mathit{\boldsymbol{\beta }}}_{\mathit{\boldsymbol{r}}}}}_{\, }\right)	0.0002076	0.0001134	0.0004153	0.0002269
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))	0.0001332	0.0001045	0.0002562	0.0002111
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)\right)	0.0001277	0.0002460	0.0002459	0.0002014
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}:\tilde{\mathit{\boldsymbol{\beta }}}(k,d))	1.6495	1.1414	1.7156	1.1308
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}:{\tilde{\mathit{\boldsymbol{\beta }}}}_{c}(k,d)\right)	1.6263	1.1373	1.6889	1.1266
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( \tilde{\mathit{\boldsymbol{\beta }}}(k,d):{\tilde{\mathit{\boldsymbol{\beta }}}}_{\mathit{\boldsymbol{c}}}(k,d)\right)	1.0430	1.1219	1.0418	1.0481
p=0.95	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\sigma }^{2},{\sigma }_{1}^{2}\right) =(0.5,0.5)		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left( {\sigma }^{2},{\sigma }_{1}^{2}\right) =(1,1)
ni	3	7	3	7
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}})	0.0005729	0.0003460	0.0011459	0.0006920
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,\left( {\tilde{{\mathit{\boldsymbol{\beta }}}_{\mathit{\boldsymbol{r}}}}}_{\, }\right)	0.0005033	0.0003044	0.0010066	0.0006088
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}(k,d))	0.0002762	0.0002760	0.0005201	0.0005272
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{EMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)\right)	0.0002578	0.0002460	0.0004927	0.0004716
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,(\tilde{\mathit{\boldsymbol{\beta }}}:\tilde{\mathit{\boldsymbol{\beta }}}(k,d))	2.0743	1.2533	2.2030	1.3125
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}:{\tilde{\mathit{\boldsymbol{\beta }}}}_{c}(k,d)\right)	1.9524	1.2369	2.0429	1.2908
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{RMSE}\,\left( \tilde{\mathit{\boldsymbol{\beta }}}(k,d):{\tilde{\mathit{\boldsymbol{\beta }}}}_{\mathit{\boldsymbol{c}}}(k,d)\right)	1.0713	1.1219	1.0556	1.1178

7. Real data analysis

We consider a data set, which is known as the Egyptian pottery data to show the behavior of the new Restricted and Unrestricted Two-Parameter Estimators. This data set arises from an extensive archaeological survey of pottery production and distribution in the ancient Egyptian city of Al-Amarna. The data consist of measurements of chemical contents (mineral elements) made on many samples of pottery using two different techniques, NAA and ICP (see Smith et al. [30] for description of techniques). The set of pottery was collected from different locations around the city. We fit the data set by linear mixed model as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{X\beta }}+\mathit{\boldsymbol{Zu}}+\mathit{\boldsymbol{\epsilon }},

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{y}}}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 159\times 1}
vector of response variables, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{X}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{Z}}}
which are regression matrix with dimensions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 159\times 6}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 159\times 25}
respectively. First, we are estimated the variance components by consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }_{1}^{2}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.5

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma }^{2}=0.5.}
Then, by calculating the eigenvalues of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{X}}{\mathit{\boldsymbol{H}}}^{-1}\mathit{\boldsymbol{X}},}
the condition number 8322860 is obtained, which indicate severe multicollinearity. We considered the stochastic linear restrictions as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{R\beta }}+\Phi ,\Phi \sim N\left( 0,{\sigma }^{2}{\mathit{\boldsymbol{I}}}_{3}\right)

 and selected 3 available data in the previous sections to the Egyptian pottery data. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{k}}_{HM}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt}}
are obtained using the iterative method introduced at the end of section 5,0.348 and 0.373 respectively. In Table 2 the estimated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MSE}
values of the estimators are obtained by replacing in the corresponding theoretical MSE equations. We can see the estimated MSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}
is less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

. Also, the estimated MSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{c}(k,d)}

is less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}}

. In general, the estimated MSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)}

is less than all estimators. So, we conclude that the stochastic restricted two parameter estimator performs better than the other estimators. Note that in the results obtained for this data, all eigenvalues of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{1}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Delta }_{2}}
are positive and the condition of Theorem 4.1 and Theorem 4.2 are true. In Figure 1 , a plot of the estimated MSE values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}
against Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
in the interval [0,2] with fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{opt~}=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.373

is drawn. Because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\beta }}
is not dependent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

, its estimated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MSE}

value is the same for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
values. It is obvious that estimated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MSE}
values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}(k,d)}
is always less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

. Altogether, it is obvious that the two parameter estimators can perform better than the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathit{\boldsymbol{\beta }}}}

in MSEM criterion under conditions.

Table 2. Estimated MSE values of the proposed estimators.

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\mathit{\boldsymbol{\beta }}}{\, }{\, }{\, }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\mathit{\boldsymbol{\beta }}}(k,d)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\mathit{\boldsymbol{\beta }}}}_{r}(k,d)
EMSE	1.216069	1.201379	0.1544953	0.1541745

Figure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1.}

The estimated mean square error values of the estimators versus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\tilde{d}}_{of~}}

8. Conclusion

In this article, we proposed the two parameter estimator and the stochastic restricted two parameter estimator to overcome the effects of the multicollinearity problem in linear mixed models. We also obtained estimates of variance parameters and then using the mean squared error matrix sense made comparisons between the proposed estimators and some other estimators. Finally, we proposed methods for estimating biasing parameters and provided a simulation study and a data example to illustrate performance of new estimators.

Disclosure statement

No potential conflict of interest was reported by the authors..

Funding

The research was supported by the Payame Noor University.

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