Statistical inference of a stochastically restricted linear mixed model

1.   Introduction

We will begin by introducing some notation before proceeding. We will write ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k, l}$ if ${\boldsymbol{\Lambda}}$ is a $k \times l$ real matrix, ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k}^{s}$ if ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k, k}$ and is symmetric, ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k}^{\succcurlyeq}$ if ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k}^{s}$ and is positive semi-definite. We will use $\boldsymbol{r}({\boldsymbol{\Lambda}})$, ${\mathscr{C}}({\boldsymbol{\Lambda}})$, ${\boldsymbol{\Lambda}}'$ and ${\boldsymbol{\Lambda}}^{+}$ as symbols to present the rank, the column space, the transpose and the Moore-Penrose generalized inverse of ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k, l}$, respectively. ${\boldsymbol{\Lambda}}^{\perp} = \mathbf{{I}}_{k}- {\boldsymbol{\Lambda}} {\boldsymbol{\Lambda}}^{+}$ stands for the orthogonal projector, where $\mathbf{{I}}_{k}\in \mathbb{R}_{k}^{s}$ is the identity matrix of order k. The symbols $\boldsymbol{i}_{+}({\boldsymbol{\Lambda}})$ and $\boldsymbol{i}_{-}({\boldsymbol{\Lambda}})$ represent positive and negative inertias of ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k}^{s}$, respectively, while $\boldsymbol{i}_{\pm}({\boldsymbol{\Lambda}})$ and $\boldsymbol{i}_{\mp}({\boldsymbol{\Lambda}})$ are used to denote both the positive and the negative inertias of ${\boldsymbol{\Lambda}} \in \mathbb{R}_{k}^{s}$ jointly. The inequality ${\boldsymbol{\Lambda}}_{1}- {\boldsymbol{\Lambda}}_{2}\succcurlyeq \mathbf{{0}}$ or ${\boldsymbol{\Lambda}}_{1}\succcurlyeq {\boldsymbol{\Lambda}}_{2}$ means ${\boldsymbol{\Lambda}}_{1}- {\boldsymbol{\Lambda}}_{2} \in \mathbb{R}_{k}^{\succcurlyeq}$ in the L$\ddot{o}$wner partial ordering for ${\boldsymbol{\Lambda}}_{1}, {\boldsymbol{\Lambda}}_{2} \in \mathbb{R}_{k}^{s}$. Similarly, when stating the other inequalities between ${\boldsymbol{\Lambda}}_{1}, {\boldsymbol{\Lambda}}_{2} \in \mathbb{R}_{k}^{s}$, ${\boldsymbol{\Lambda}}_{1}- {\boldsymbol{\Lambda}}_{2}\succ \mathbf{{0}}$ (${\boldsymbol{\Lambda}}_{1}\succ {\boldsymbol{\Lambda}}_{2}$), ${\boldsymbol{\Lambda}}_{1}- {\boldsymbol{\Lambda}}_{2}\prec \mathbf{{0}}$ (${\boldsymbol{\Lambda}}_{1}\prec {\boldsymbol{\Lambda}}_{2}$), ${\boldsymbol{\Lambda}}_{1}- {\boldsymbol{\Lambda}}_{2}\preccurlyeq \mathbf{{0}}$ (${\boldsymbol{\Lambda}}_{1}\preccurlyeq {\boldsymbol{\Lambda}}_{2}$) are used if the difference ${\boldsymbol{\Lambda}}_{1}- {\boldsymbol{\Lambda}}_{2}$ is positive definite, negative definite and negative semi-definite matrices, respectively. We will write ${\rm{E}}({\boldsymbol{\lambda}})$ for the expectation vector and ${\rm{D}}({\boldsymbol{\lambda}}) = {\rm{cov}}({\boldsymbol{\lambda}}, {\boldsymbol{\lambda}})$ for the dispersion matrix of a random vector ${\boldsymbol{\lambda}}\in \mathbb{R}_{k, 1}$, where ${\rm{cov}}({\boldsymbol{\lambda}}, {\boldsymbol{\lambda}})$ denotes the covariance matrix of ${\boldsymbol{\lambda}}$.

A linear mixed model (${\rm{LMM}}$) is a variant of linear regression model that combines fixed and random effects in the same analysis, allowing for more flexibility in model fitting. In some statistical problems, besides the sample information of ${\rm{LMM}}$s, there may exist additional information, usually produced as prior information for the models. This prior information can be expressed as either a linear stochastic restriction or a linear exact restriction on unknown parameters, and it is usually added to the assumptions of models. Linear exact restrictions occur when there is a specific linear hypothesis being tested or when exact knowledge is available among certain parameters. On the other hand, linear stochastic restrictions arise when prior information is derived from previous investigations or established long-term relationships with relevant studies; see, e.g., ^[1,2].

An ${\rm{LMM}}$ with a linear stochastic restriction, also known as a stochastically restricted ${\rm{LMM}}$ (${\rm{SRLMM}}$), can be given as follows:

where $\mathbf{{y}} \in \mathbb{R}_{n, 1}$ is a vector of responses, $\mathbf{{X}}\in \mathbb{R}_{n, k}$, $\mathbf{{Z}}\in\mathbb{R}_{n, p}$, $\mathbf{{S}}\in \mathbb{R}_{m, k}$ are known matrices of arbitrary ranks and $\mathbf{{s}}\in \mathbb{R}_{m, 1}$ is a known vector, ${\boldsymbol{\alpha}}\in \mathbb{R}_{k, 1}$ is a parameter vector of fixed effects, ${\boldsymbol{\gamma}} \in \mathbb{R}_{p, 1}$ is a vector of random effects, ${\boldsymbol{\varepsilon}} \in \mathbb{R}_{n, 1}$ and $\mathbf{{e}} \in \mathbb{R}_{m, 1}$ are vectors of random errors, ${\boldsymbol{\Lambda}} \in \mathbb{R}_{n+p+m}^{\succcurlyeq}$ of arbitrary ranks and its entries ${\boldsymbol{\Lambda}}_{i}$ are known, $i = 1, 2, 3$ and ${\boldsymbol{\sigma}}^{2}$ is a positive unknown parameter.

In the present study, we consider the ${\rm{SRLMM}}$ in (1.1). We derive the best linear unbiased predictors (${\rm{BLUP}}$s) of a unified form of all unknown parameters in the ${\rm{SRLMM}}$ utilizing some quadratic matrix optimization methods, which include block matrix inertias and ranks. Furthermore, we discuss some of the ${\rm{BLUP}}$s' basic characteristics. We specifically address the comparison problem between any predictor and the ${\rm{BLUP}}$ for all unknown parameters in a unified form under the ${\rm{SRLMM}}$ in the sense of the mean squared error matrix (${\rm{MSEM}}$) criterion. It is well-known that the ${\rm{MSEM}}$ of any predictor or estimator $\widetilde {\boldsymbol{\lambda}}$ of ${\boldsymbol{\lambda}}$ in a linear regression model is defined as the matrix

To compare two given predictors or estimators $\widetilde {\boldsymbol{\lambda}}_{1}$ and $\widetilde {\boldsymbol{\lambda}}_{2}$, the difference

is written. In this case, $\widetilde {\boldsymbol{\lambda}}_{2}$ is said to be superior to $\widetilde {\boldsymbol{\lambda}}_{1}$ with respect to the ${\rm{MSEM}}$ criterion iff $\Delta(\widetilde {\boldsymbol{\lambda}}_{1}, \widetilde {\boldsymbol{\lambda}}_{2})\succcurlyeq \mathbf{{0}}$ holds. Using the methodology of block matrix inertias and ranks, a number of equalities and inequalities for the comparisons of ${\rm{MSEM}}$s of two predictors, with one being the ${\rm{BLUP}}$, of a unified form of all unknown parameters are established by considering the difference in (1.4). Such comparisons allow us to evaluate whether two predictors, one of which is ${\rm{BLUP}}$, of the same unknown parameter vector are more effective than each other. We also derive comparison results for special cases of a unified form of all unknown parameters under the ${\rm{SRLMM}}$. Furthermore, the results are reduced to both a constrained linear mixed model (${\rm{CLMM}}$), i.e., an ${\rm{LMM}}$ with a linear exact restriction, and an unconstrained ${\rm{LMM}}$ (${\rm{ULMM}}$). We use a numerical example to explain the theoretical results. There have been numerous previous and recent studies in the literature that focus on an exact restriction on unknown parameters under ${\rm{LMM}}$s and linear models from different perspectives; see, e.g., ^{[3,4,5,6,7,8,9,10,11,12,13,14]} and the references therein. There has also been considerable attention on stochastic restriction on unknown parameters in a linear regression model from different approaches; see, e.g., ^{[1,2,15,16,17,18]}, among others. We may also refer to ^{[19,20,21,22,23,24,25,26,27,28]}, among others, for the corresponding studies in which similar approaches were conducted for the comparisons of predictors or estimators by utilizing the methodology of block matrix inertias and ranks. It is worth noting that studies on stochastic or exact restrictions are not limited to the context of linear models and in the field of statistics. The stochastic or exact restrictions on linear or nonlinear equations have been widely investigated from various aspects in a variety of fields, including biostatistics, educational measurement, sociology, finance, biology, chemistry, mechanics and economics, see, e.g., ^{[29,30,31,32,33,34]}.

The development of various potential predictors (or, also potential estimators) of unknown parameter vectors in linear regression models is one of the main goals of theoretical and practical studies. In order to define predictors of unknown parameter vectors in the model, various types of optimality criteria have been utilized in mathematics. The unbiasedness of predictors of unknown parameter vectors according to the parameter spaces in the model is one of the crucial characteristics among others. When there are numerous unbiased predictors for the same parameter space, finding the one with the smallest dispersion matrix becomes significant. The ${\rm{BLUP}}$, by its definition, corresponds to these two requirements. Thus, due to the minimizing property of the ${\rm{BLUP}}$'s dispersion matrix, the ${\rm{BLUP}}$ has a significant role to play in statistical inference theory and is frequently used as the foundation for evaluating the effectiveness of various predictors. As stated in the next section, the dispersion matrix of the ${\rm{BLUP}}$ is equal to its ${\rm{MSEM}}$. In this situation, in the context of the ${\rm{SRLMM}}$, comparing any predictor with the ${\rm{BLUP}}$ for a unified form of all unknown parameters using the ${\rm{MSEM}}$ criterion allows us to determine the optimality of the compared predictor. As it is known, one of the main criteria for determining the superiority of two different predictors of the unknown parameter vectors in the model is the ${\rm{MSEM}}$ criterion, which is widely used to measure predictors accurately.

The expressions, which are obtained from the comparison issues of the predictors, include various complicated matrix operations, including the use of Moore-Penrose generalized inverses. Some of the methods to simplify expressions composed by matrices and their Moore-Penrose generalized inverses are based on conventional operations on matrices, as well as some known equalities for inverses or Moore-Penrose generalized inverses of matrices. These methods' efficacy is quite limited, and the related processes are quite tedious. However, the methodology of block matrix inertias and ranks offers an effective approach to simplify such complex matrix expressions. It allows for the representation of results as simple inertia and rank equalities. The theory of matrix inertias and ranks has been developed as an efficient methodology for the simplification of such complex matrix expressions. In linear algebra, the inertia and rank of a matrix are fundamental quantities that are straightforward to understand and calculate. Using the theory of inertias and ranks of the matrix, various inequalities and equalities between the ${\rm{MSEM}}$s of predictors correspond to the comparison of quantities derived from inertias and ranks of matrices.

The paper presents novel theoretical techniques and innovative approaches for comparing any predictor and the ${\rm{BLUP}}$ of the same general vector of unknown parameters under the ${\rm{SRLMM}}$. The comparisons are based on the ${\rm{MSEM}}$ criterion, and the methodology of block matrix inertias and ranks is employed to establish exact formulas for these comparisons. It is important to note that we do not make any distributional assumptions for the random vectors in the models, except for assuming the existence of first and second moments. Additionally, no restrictions are imposed on the ranks of given matrices. Furthermore, there are no specific requirements concerning the patterns of the submatrices of ${\boldsymbol{\Lambda}}$ in (1.2). In statistical practice, if we meet the cases where ${\boldsymbol{\Lambda}}$ is unknown, then the estimators of submatrices of ${\boldsymbol{\Lambda}}$ can be used. Alternatively, if ${\boldsymbol{\Lambda}}$ is given with some parametric forms and some specific patterns, then the estimators obtained from the observed data can be substituted. These are other types of inference work, which we do not consider in this article.

Lastly, we list a few lemmas that are essential for the subsequent sections of the paper, contributing to the overall theoretical framework developed for comparing predictors under the ${\rm{SRLMM}}$.

Lemma 1.1 (^[35]). Let ${\boldsymbol{\Lambda}}_{1}$, ${\boldsymbol{\Lambda}}_{2}$ $\in \mathbb{R}_{k, l}$, or, let ${\boldsymbol{\Lambda}}_{1}$, ${\boldsymbol{\Lambda}}_{2}$ $\in \mathbb{R}_{k}^{s}$. Then,

Lemma 1.2 (^[35]). Let ${\boldsymbol{\Lambda}}_{1} \in \mathbb{R}_{k}^{s}$, ${\boldsymbol{\Lambda}}_{2} \in \mathbb{R}_{k, l}$, ${\boldsymbol{\Lambda}}_{3} \in \mathbb{R}_{l}^{s}$ and $a \in \mathbb{R}$. Then,

Lemma 1.3 (^[36]). Let ${\boldsymbol{\Lambda}}_{1} \in \mathbb{R}_{k, l}$ and ${\boldsymbol{\Lambda}}_{2} \in \mathbb{R}_{m, l}$ be given matrices, and let $\mathbf{{H}} \in \mathbb{R}_{k}^{\succcurlyeq}$. Suppose that there exists $\mathbf{{X}} \in \mathbb{R}_{m, k}$ such that $\mathbf{{X}} {\boldsymbol{\Lambda}}_{1} = {\boldsymbol{\Lambda}}_{2}$. Then the maximal positive inertia of $\mathbf{{X}} \mathbf{{H}} \mathbf{{X}}'- \mathbf{{Y}} \mathbf{{H}} \mathbf{{Y}}'$ s.t. all solutions of $\mathbf{{Y}} {\boldsymbol{\Lambda}}_{1} = {\boldsymbol{\Lambda}}_{2}$ is

Hence, a solution $\mathbf{{X}}$ of $\mathbf{{X}} {\boldsymbol{\Lambda}}_{1} = {\boldsymbol{\Lambda}}_{2}$ exists such that $\mathbf{{X}} \mathbf{{H}} \mathbf{{X}}'\preccurlyeq \mathbf{{Y}} \mathbf{{H}} \mathbf{{Y}}'$ holds for all solutions of $\mathbf{{Y}} {\boldsymbol{\Lambda}}_{1} = {\boldsymbol{\Lambda}}_{2}$ $\Leftrightarrow$ both the equations $\mathbf{{X}} {\boldsymbol{\Lambda}}_{1} = {\boldsymbol{\Lambda}}_{2}$ and $\mathbf{{X}} \mathbf{{H}} {\boldsymbol{\Lambda}}_{1}^{\perp} = \mathbf{{0}}$ are satisfied by $\mathbf{{X}}$.

2.   BLUPs under SRLMMs

Let us consider the ${\rm{SRLMM}}$ in (1.1). By unifying two given equation parts in (1.1), we obtain

Through the unifying operation that we use in (2.1), which is a common procedure for approaching two or more equations, the ${\rm{SRLMM}}$ in (1.1) is converted to the implicitly linear stochastically restricted ${\rm{LMM}}$ in (2.1). We can take into consideration the following general vector to produce conclusions on predictors of all unknown vectors under ${\mathcal{S}}$:

where $\mathbf{{K}}\in \mathbb{R}_{t, k}$, $\mathbf{{L}} \in \mathbb{R}_{t, p}$, $\mathbf{{M}}_{1} \in \mathbb{R}_{t, n}$ and $\mathbf{{M}}_{2} \in \mathbb{R}_{t, m}$ with $\mathbf{{M}} = \begin{bmatrix} \mathbf{{M}}_{1}, & \mathbf{{M}}_{2} \end{bmatrix}$ are given matrices of arbitrary ranks. Then, from (1.2), (2.1) and (2.2),

where ${\boldsymbol{\Phi}} = \begin{bmatrix} \mathbf{{Z}}_{s}, & \mathbf{{I}}_{n+m} \end{bmatrix}$ and ${\boldsymbol{\Upsilon}} = \begin{bmatrix} \mathbf{{L}}, & \mathbf{{M}} \end{bmatrix}$. Further, in this study, the model ${\mathcal{S}}$ is assumed to be consistent, i.e., $\mathbf{{y}}_{s} \in {\mathscr{C}}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}$ holds with probability 1; see, e.g., ^[37].

We introduce the following definition of the predictability of ${\boldsymbol{\lambda}}$ and its subcases; see, e.g., ^[38,39]. Later, the definition of ${\rm{BLUP}}$ of ${\boldsymbol{\lambda}}$ under ${\mathcal{S}}$ is given; see, e.g., ^[40,41].

Definition 2.1. Let consider ${\mathcal{S}}$ in (2.1) and ${\boldsymbol{\lambda}}$ in (2.2). Then,

1) If a matrix $\mathbf{{F}} \in \mathbb{R}_{t, (n+m)}$ exists with ${\rm{E}}(\mathbf{{F}} \mathbf{{y}}_{s}- {\boldsymbol{\lambda}}) = \mathbf{{0}}$, then ${\boldsymbol{\lambda}}$ is defined to be predictable, that is, ${\boldsymbol{\lambda}}$ is predictable $\Leftrightarrow$ ${\mathscr{C}}(\mathbf{{K}}')\subseteq {\mathscr{C}}(\mathbf{{X}}_{s}')$ $\Leftrightarrow$ $\mathbf{{K}} {\boldsymbol{\alpha}}$ is estimable.

2) $\mathbf{{X}}_{s} {\boldsymbol{\alpha}}+ \mathbf{{Z}}_{s} {\boldsymbol{\gamma}}+ {\boldsymbol{\varepsilon}}_{s}$ is always predictable and $\mathbf{{X}}_{s} {\boldsymbol{\alpha}}$ is always estimable.

3) ${\boldsymbol{\alpha}}$ is estimable $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{X}}_{s}) = k$.

4) ${\boldsymbol{\gamma}}$, ${\boldsymbol{\varepsilon}}$ and $\mathbf{{e}}$ are always predictable.

Definition 2.2. Suppose that ${\boldsymbol{\lambda}}$ in (2.2) is predictable under ${\mathcal{S}}$ in (2.1). $\mathbf{{F}} \mathbf{{y}}_{s}$ is said to be the ${\rm{BLUP}}$ of ${\boldsymbol{\lambda}}$, represented by $\mathbf{{F}} \mathbf{{y}}_{s} = \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}$, if there exists $\mathbf{{F}} \mathbf{{y}}_{s}$ such that

holds in the L$\ddot{o}$wner partial ordering. If $\mathbf{{L}} = \mathbf{{0}}$ and $\mathbf{{M}} = \mathbf{{0}}$ in ${\boldsymbol{\lambda}}$, $\mathbf{{F}} \mathbf{{y}}_{s}$ is the well-known the best linear unbiased estimator $({\rm{BLUE}})$ of $\mathbf{{K}} {\boldsymbol{\alpha}}$, represented by $\widehat{ \mathbf{{K}} {\boldsymbol{\alpha}}}_{ {\rm{BLUE}}}$.

According to the definition of the ${\rm{MSEM}}$ given in (1.3), we can give the following lemma, see; ^[42,43].

Lemma 2.1.. Let $\widetilde {\boldsymbol{\lambda}}$ be a predictor for a general vector of all unknown vectors ${\boldsymbol{\lambda}}$ in an ${\rm{LMM}}$. Then,

where ${\rm{bias}}(\widetilde {\boldsymbol{\lambda}}) = {\rm{E}}(\widetilde {\boldsymbol{\lambda}}- {\boldsymbol{\lambda}})$. If $\widehat {\boldsymbol{\lambda}}$ is an estimator for a general vector of all unknown parameters ${\boldsymbol{\lambda}}$ in an ${\rm{LMM}}$, then,

where ${\rm{bias}}(\widehat {\boldsymbol{\lambda}}) = {\rm{E}}(\widehat {\boldsymbol{\lambda}})- {\boldsymbol{\lambda}}$.

The fundamental ${\rm{BLUP}}$ equation and associated properties are summarized in the following lemma; see ^[25], for different approaches; see also, ^[37,41,44].

Lemma 2.2. Suppose that ${\boldsymbol{\lambda}}$ in (2.2) is predictable under ${\mathcal{S}}$ in (2.1). Consider any two unbiased linear predictors $\mathbf{{F}} \mathbf{{y}}_{s}$ and $\mathbf{{G}} \mathbf{{y}}_{s}$ for ${\boldsymbol{\lambda}}$. Then,

is the maximal positive inertia of ${\rm{D}}(\mathbf{{F}} \mathbf{{y}}_{s}- {\boldsymbol{\lambda}})- {\rm{D}}(\mathbf{{G}} \mathbf{{y}}_{s}- {\boldsymbol{\lambda}})$ s.t. $\mathbf{{G}} \mathbf{{X}}_{s} = \mathbf{{K}}$. Hence, ${\rm{D}}(\mathbf{{F}} \mathbf{{y}}_{s}- {\boldsymbol{\lambda}}) = \hbox {min s.t. } {\rm{E}}(\mathbf{{F}} \mathbf{{y}}_{s}- {\boldsymbol{\lambda}}) = \mathbf{{0}} \Leftrightarrow \mathbf{{F}} \mathbf{{y}}_{s} = \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}$

This equation in (2.6) is consistent and the ${\rm{BLUP}}$ of ${\boldsymbol{\lambda}}$ under ${\mathcal{S}}$ can be written as follows by considering the general solution of this equation:

where $\mathbf{{J}} = \begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \mathbf{{X}}_{s}^{\perp}\end{bmatrix}$ and $\mathbf{{U}} \in \mathbb{R}_{t, (n+m)}$ is an arbitrary matrix. Further,

and the ${\rm{MSEM}}$ of $\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}$ is

In particular,

1) ${\mathscr{C}}(\mathbf{{J}}) = \mathbb{R}_{(n+m), 1}$ $\Leftrightarrow$ $\mathbf{{F}}$ is unique and ${\mathcal{S}}$ is consistent $\Leftrightarrow$ $\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}$ is unique.

2) The following equalities hold:

Proof of Lemma 2.2. For an unbiased linear predictor $\mathbf{{F}} \mathbf{{y}}_{s}$ of ${\boldsymbol{\lambda}}$ in ${\mathcal{S}}$,

are written. For the other unbiased predictor $\mathbf{{G}} \mathbf{{y}}_{s}$ of ${\boldsymbol{\lambda}}$, the similar expressions can be written as in (2.13) by putting $\mathbf{{G}}$ instead of $\mathbf{{F}}$. Finding solution $\mathbf{{F}}$ of the consistent equation $\mathbf{{F}} \mathbf{{X}}_{s} = \mathbf{{K}}$ such that

corresponds to find the ${\rm{BLUP}}$ of ${\boldsymbol{\lambda}}$ under ${\mathcal{S}}$. According to Lemma 1.3, (2.14) is a typical constrained quadratic matrix-valued function optimization problem in the L$\ddot{o}$wner partial ordering. Applying Lemma 1.3 to (2.14), (2.5) is obtained. Using Lemma 1.3, we obtain the fundamental ${\rm{BLUP}}$ equation of ${\boldsymbol{\lambda}}$ in (2.6). The consistency of the equation in (2.6) is seen from the following column space inclusions:

The well-known general solution of (2.6) s.t. $\mathbf{{F}}$ can be written in the parametric form as in (2.7). (2.8) and (2.9) are directly seen from (2.3) and (2.7). ${\rm{D}}({\boldsymbol{\lambda}}-\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$ is written as

By setting the equalities in (2.3), (2.8) and (2.9) into (2.16), (2.10) is obtained. (2.4) and (2.16) give (2.11) since ${\rm{bias}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}) = \mathbf{{0}}$. Item 1 follows from (2.7). The expressions in (2.12) are well-known results; see [45,Lemma 2.1 (a)]. □

We note that the fundamental equations of ${\rm{BLUP}}$s and ${\rm{BLUE}}$s for unknown vectors in ${\boldsymbol{\lambda}}$ and their related results can be derived from 5, by special choices of the matrices $\mathbf{{K}}$, $\mathbf{{L}}$ and $\mathbf{{M}}$.

3.   Comparisons under SRLMM

The results of the relationships between the ${\rm{MSEM}}$ of any predictor for ${\boldsymbol{\lambda}}$ and the ${\rm{MSEM}}$ of the ${\rm{BLUP}}$ of ${\boldsymbol{\lambda}}$ under ${\mathcal{S}}$ are collected in the theorem given below.

Theorem 3.1. Suppose that ${\boldsymbol{\lambda}}$ in (2.2) is predictable under ${\mathcal{S}}$ in (2.1). Let $\widetilde {\boldsymbol{\lambda}}$ be any predictor (unbiased or biased) for ${\boldsymbol{\lambda}}$ under ${\mathcal{S}}$. Denote

and

Then,

In consequence,

1) $\Delta(\widetilde {\boldsymbol{\lambda}}, \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})\succcurlyeq \mathbf{{0}}$, i.e., ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}})\succcurlyeq {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$$\Leftrightarrow \boldsymbol{i}_{+}(\mathbf{{E}}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}$.

2) $\Delta(\widetilde {\boldsymbol{\lambda}}, \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})\succ \mathbf{{0}}$, i.e., ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}})\succ {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$ $\Leftrightarrow \boldsymbol{i}_{-}(\mathbf{{E}}) = \boldsymbol{r}(\mathbf{{X}}_{s})+t$.

3) $\Delta(\widetilde {\boldsymbol{\lambda}}, \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}) \preccurlyeq \mathbf{{0}}$, i.e., ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}) \preccurlyeq {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$ $\Leftrightarrow \boldsymbol{i}_{-}(\mathbf{{E}}) = \boldsymbol{r}(\mathbf{{X}}_{s})$.

4) $\Delta(\widetilde {\boldsymbol{\lambda}}, \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}) \prec \mathbf{{0}}$, i.e., ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}) \prec {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$ $\Leftrightarrow \boldsymbol{i}_{+}(\mathbf{{E}}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+t$.

5) $\Delta(\widetilde {\boldsymbol{\lambda}}, \widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}}) = \mathbf{{0}}$, i.e., ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}) = {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$$\Leftrightarrow \boldsymbol{r}(\mathbf{{E}}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}_{s})$.

Proof. Let say $\mathbf{{C}} = {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}})$ and apply (1.11) to the difference between ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}})$ and ${\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}}_{ {\rm{BLUP}}})$ in (2.11). Then, we obtain

where $\mathbf{{D}} = \begin{bmatrix} \mathbf{{K}}, & {\boldsymbol{\Upsilon}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \mathbf{{X}}_{s}^{\perp}\end{bmatrix}$ and $\mathbf{{J}} = \begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \mathbf{{X}}_{s}^{\perp}\end{bmatrix}$. By using column space equalities and inclusions in (2.12) and (2.15), reapplying (1.11) to (3.4) and also using (1.6) and (1.8) with the elementary block matrix operations, the expression in (3.4) is equivalently written as

By setting $\mathbf{{D}}$ and $\mathbf{{J}}$ into (3.5) and also using (1.7)–(1.9) and (2.12) with the elementary block matrix operations,

is obtained. From (1.10),

are written and then, after setting $\mathbf{{C}} = {\rm{MSEM}}(\widetilde {\boldsymbol{\lambda}})$ in the first matrix in (3.6), (3.1) and (3.2) are obtained from (3.6) and (3.7). According to (1.5), adding the equalities in (3.1) and (3.2) yields (3.3). Applying Lemma 1.1 to (3.1)–(3.3) yields 1–5. □

Corollary 3.1. Let ${\mathcal{S}}$ and ${\boldsymbol{\lambda}}$ be as given in (2.1) and (2.2), respectively. Suppose that $\widetilde {\boldsymbol{\gamma}}$, $\widetilde {\boldsymbol{\varepsilon}}$ and $\widetilde {\bf{e}}$ are predictors for ${\boldsymbol{\gamma}}$, ${\boldsymbol{\varepsilon}}$ and $\mathbf{{e}}$, respectively. Let say ${\boldsymbol{\mu}} = \mathbf{{X}}_{s} {\boldsymbol{\alpha}}$ and $\widehat{ {\boldsymbol{\mu}}}$ be any estimator for ${\boldsymbol{\mu}}$ under ${\mathcal{S}}$. Denote

where $\breve {\bf{I}}_{p} = \begin{bmatrix} \mathbf{{I}}_{p}, & \mathbf{{0}}, & \mathbf{{0}} \end{bmatrix}$, $\breve {\bf{I}}_{n} = \begin{bmatrix} \mathbf{{0}}, & \mathbf{{I}}_{n}, & \mathbf{{0}} \end{bmatrix}$ and $\breve {\bf{I}}_{m} = \begin{bmatrix} \mathbf{{0}}, & \mathbf{{0}}, & \mathbf{{I}}_{m}\end{bmatrix}$. Then,

1) ${\rm{MSEM}}(\widehat{ {\boldsymbol{\mu}}}) \succcurlyeq {\rm{MSEM}}(\widehat {\boldsymbol{\mu}}_{ {\rm{BLUE}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{1}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}$.

2) ${\rm{MSEM}}(\widehat{ {\boldsymbol{\mu}}})\succ {\rm{MSEM}}(\widehat {\boldsymbol{\mu}}_{ {\rm{BLUE}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{1}) = \boldsymbol{r}(\mathbf{{X}}_{s})+n+m$.

3) ${\rm{MSEM}}(\widehat{ {\boldsymbol{\mu}}}) \preccurlyeq {\rm{MSEM}}(\widehat {\boldsymbol{\mu}}_{ {\rm{BLUE}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{1}) = \boldsymbol{r}(\mathbf{{X}}_{s}).$

4) ${\rm{MSEM}}(\widehat{ {\boldsymbol{\mu}}}) \prec {\rm{MSEM}}(\widehat {\boldsymbol{\mu}}_{ {\rm{BLUE}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{1}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+n+m$.

5) ${\rm{MSEM}}(\widehat{ {\boldsymbol{\mu}}}) = {\rm{MSEM}}(\widehat {\boldsymbol{\mu}}_{ {\rm{BLUE}}})$ $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{E}}_{1}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}_{s}).$

6) ${\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}) \succcurlyeq {\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{2}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}$.

7) ${\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}})\succ {\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{2}) = \boldsymbol{r}(\mathbf{{X}}_{s})+p$.

8) ${\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}) \preccurlyeq {\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{2}) = \boldsymbol{r}(\mathbf{{X}}_{s}).$

9) ${\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}})\prec {\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{2}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+p$.

10) ${\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}) = {\rm{MSEM}}(\widetilde {\boldsymbol{\gamma}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{E}}_{2}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}_{s}).$

11) ${\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}) \succcurlyeq {\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{3}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}$.

12) ${\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}})\succ {\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{3}) = \boldsymbol{r}(\mathbf{{X}}_{s})+n$.

13) ${\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}) \preccurlyeq {\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{3}) = \boldsymbol{r}(\mathbf{{X}}_{s}).$

14) ${\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}) \prec {\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{3}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+n$.

15) ${\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}) = {\rm{MSEM}}(\widetilde {\boldsymbol{\varepsilon}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{E}}_{3}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}_{s}).$

16) ${\rm{MSEM}}(\widetilde {\bf{e}})\succcurlyeq {\rm{MSEM}}(\widetilde {\bf{e}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{4}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}$.

17) ${\rm{MSEM}}(\widetilde {\bf{e}})\succ {\rm{MSEM}}(\widetilde {\bf{e}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{4}) = \boldsymbol{r}(\mathbf{{X}}_{s})+m$.

18) ${\rm{MSEM}}(\widetilde {\bf{e}}) \preccurlyeq {\rm{MSEM}}(\widetilde {\bf{e}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{4}) = \boldsymbol{r}(\mathbf{{X}}_{s}).$

19) ${\rm{MSEM}}(\widetilde {\bf{e}})\prec {\rm{MSEM}}(\widetilde {\bf{e}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{4}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+m$.

20) ${\rm{MSEM}}(\widetilde {\bf{e}}) = {\rm{MSEM}}(\widetilde {\bf{e}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{E}}_{4}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}} {\boldsymbol{\Phi}}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}_{s}).$

4.   Comparisons under CLMMs and ULMMs

Now, let us consider the ${\rm{LMM}}$ in (1.1) with an exact restriction $\mathbf{{S}} {\boldsymbol{\alpha}} = \mathbf{{s}}$. Then, the model ${\mathcal{S}}$ in (2.1) corresponds the following ${\rm{CLMM}}$:

where $\mathbf{{y}}_{s}$, $\mathbf{{X}}_{s}$ and $\mathbf{{Z}}_{s}$ are given as in (2.1) and ${\boldsymbol{\varepsilon}}_{r} = \begin{bmatrix} {\boldsymbol{\varepsilon}} \\ \mathbf{{0}} \end{bmatrix}$. ${\boldsymbol{\lambda}}$ in (2.2) corresponds the vector

In this case, the assumptions under ${\mathcal{R}}$ are written as

where ${\boldsymbol{\Lambda}}_{r} = \begin{bmatrix} {\boldsymbol{\Lambda}}_{1} & \mathbf{{0}} \\ \mathbf{{0}} & \mathbf{{0}} \end{bmatrix}$.

In the theorem given below, the results on the relationships between the ${\rm{MSEM}}$ of any predictor for $\mathbf{{r}}$ and the ${\rm{MSEM}}$ of the ${\rm{BLUP}}$ of $\mathbf{{r}}$ under ${\mathcal{R}}$ are collected. These results are obtained from Theorem 3.1.

Theorem 4.1. Suppose that $\mathbf{{r}}$ in (4.2) is predictable under ${\mathcal{R}}$ in (4.1). Let $\widetilde {\bf{r}}$ be any predictor (unbiased or biased) for $\mathbf{{r}}$ under ${\mathcal{R}}$. Denote

Then, the following results hold.

1) ${\rm{MSEM}}(\widetilde {\bf{r}}) \succcurlyeq {\rm{MSEM}}(\widetilde {\bf{r}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{r}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}}_{r} {\boldsymbol{\Phi}}' \end{bmatrix}$.

2) ${\rm{MSEM}}(\widetilde {\bf{r}}) \succ {\rm{MSEM}}(\widetilde {\bf{r}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{r}) = \boldsymbol{r}(\mathbf{{X}}_{s})+t$.

3) ${\rm{MSEM}}(\widetilde {\bf{r}}) \preccurlyeq {\rm{MSEM}}(\widetilde {\bf{r}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{r}) = \boldsymbol{r}(\mathbf{{X}}_{s}).$

4) ${\rm{MSEM}}(\widetilde {\bf{r}}) \prec {\rm{MSEM}}(\widetilde {\bf{r}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{r}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}}_{r} {\boldsymbol{\Phi}}' \end{bmatrix}+t$.

5) ${\rm{MSEM}}(\widetilde {\bf{r}}) = {\rm{MSEM}}(\widetilde {\bf{r}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{E}}_{r}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_{s}, & {\boldsymbol{\Phi}} {\boldsymbol{\Lambda}}_{r} {\boldsymbol{\Phi}}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}_{s}).$

Now, let us consider the ${\rm{LMM}}$ in (1.1) without any restrictions on parameters and show this model ${\mathcal{U}}$, i.e.,

and ${\boldsymbol{\lambda}}$ in (2.2) corresponds the vector

In this case,

Theorem 4.2. Suppose that $\mathbf{{u}}$ in (4.4) is predictable under ${\mathcal{U}}$ in (4.3). Let $\widetilde {\bf{u}}$ be any predictor (unbiased or biased) for $\mathbf{{u}}$ under ${\mathcal{U}}$. Denote

where ${\boldsymbol{\Phi}}_{1} = \begin{bmatrix} \mathbf{{Z}}, & \mathbf{{I}}_{n} \end{bmatrix}$ and ${\boldsymbol{\Upsilon}}_{1} = \begin{bmatrix} \mathbf{{L}}, & \mathbf{{M}}_{1} \end{bmatrix}$. Then, the following results hold.

1) ${\rm{MSEM}}(\widetilde {\bf{u}}) \succcurlyeq {\rm{MSEM}}(\widetilde {\bf{u}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{u}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}, & {\boldsymbol{\Phi}}_{1} {\boldsymbol{\Lambda}}_{1} {\boldsymbol{\Phi}}_{1}' \end{bmatrix}$.

2) ${\rm{MSEM}}(\widetilde {\bf{u}}) \succ {\rm{MSEM}}(\widetilde {\bf{u}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{u}) = \boldsymbol{r}(\mathbf{{X}})+t$.

3) ${\rm{MSEM}}(\widetilde {\bf{u}}) \preccurlyeq {\rm{MSEM}}(\widetilde {\bf{u}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{-}(\mathbf{{E}}_{u}) = \boldsymbol{r}(\mathbf{{X}}).$

4) ${\rm{MSEM}}(\widetilde {\bf{u}}) \prec {\rm{MSEM}}(\widetilde {\bf{u}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{i}_{+}(\mathbf{{E}}_{u}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}, & {\boldsymbol{\Phi}}_{1} {\boldsymbol{\Lambda}}_{1} {\boldsymbol{\Phi}}_{1}' \end{bmatrix}+t$.

5) ${\rm{MSEM}}(\widetilde {\bf{u}}) = {\rm{MSEM}}(\widetilde {\bf{u}}_{ {\rm{BLUP}}})$ $\Leftrightarrow$ $\boldsymbol{r}(\mathbf{{E}}_{u}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}, & {\boldsymbol{\Phi}}_{1} {\boldsymbol{\Lambda}}_{1} {\boldsymbol{\Phi}}_{1}' \end{bmatrix}+ \boldsymbol{r}(\mathbf{{X}}).$

As is well-known, two types of commonly used estimators under a general linear model are the ${\rm{BLUE}}$s and the ordinary least squares estimators (${\rm{OLSE}}$s). Both of these estimators have a variety of simple and remarkable properties. Therefore, they have attracted statisticians' attention throughout the historical development of regression theory, and numerous results regarding the ${\rm{BLUE}}$s and the ${\rm{OLSE}}$s have been established. Since these estimators are defined by different optimality criteria and thereby their expressions and properties are not necessarily the same, it is natural to seek possible connections between them. As a subject area within the regression analysis, the relationships between the ${\rm{BLUE}}$s and ${\rm{OLSE}}$s have been widely considered in the statistical literature and various identifying conditions for their equivalence or comparisons have been obtained. Based on these explanations, to further clarify the results obtained above, since both the ${\rm{BLUE}}$ and the ${\rm{OLSE}}$ of $\mathbf{{X}}_{s} {\boldsymbol{\alpha}}$, denoted as $\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{OLSE}}}$, are well-known estimators, we present the following result on the relationship between them.

Let us consider the ${\rm{LMM}}$ in (1.1) by setting $\mathbf{{Z}} = \mathbf{{0}}$ with an exact restriction $\mathbf{{S}} {\boldsymbol{\alpha}} = \mathbf{{s}}$. Then, the model ${\mathcal{S}}$ in (2.1) corresponds the following linear model:

where $\mathbf{{y}}_{s}$ and $\mathbf{{X}}_{s}$ are given as in (2.1) and ${\boldsymbol{\varepsilon}}_{r}$ is given as in (4.1). By taking ${\boldsymbol{\lambda}} = \mathbf{{X}}_{s} {\boldsymbol{\alpha}}$ in (2.2), the matrix $\mathbf{{E}}$ in Theorem 1 corresponds

where ${\boldsymbol{\Lambda}}_{r} = \begin{bmatrix} {\boldsymbol{\Lambda}}_{11} & \mathbf{{0}} \\ \mathbf{{0}} & \mathbf{{0}} \end{bmatrix}$ and ${\rm{D}}({\boldsymbol{\varepsilon}}) = {\boldsymbol{\Lambda}}_{11}$. Then,

always holds, i.e., ${\rm{MSEM}}(\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{OLSE}}}) \succcurlyeq {\rm{MSEM}}(\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{BLUE}}})$. This inequality is already a well-known result in statistical theory.

We now use Example 4.1.8 from Section 4.1 of ^[46] to illustrate our theoretical findings in response to the referees' advice. Consider the linear model $\mathbf{{y}} = \mathbf{{X}} {\boldsymbol{\alpha}}+ {\boldsymbol{\varepsilon}}$ with

where $\mathbf{{1}}\in\mathbb{R}_{10, 1}$ and $\mathbf{{0}}\in\mathbb{R}_{10, 1}$. Let consider

as in ^[2]. This designed set-up is typical in agricultural experiments where several treatments are applied to various blocks of land. The experiment is often conducted to assess the differential impact of the treatments. Here, the parameter $\mu$ represents a general effect that is present in all the observations, and the parameters $\beta_{1}$ and $\beta_{2}$ represent the respective effects of two blocks, and the parameters $\tau_{1}$ and $\tau_{2}$ represent the respective effects of two treatments. Suppose that

where $\mathbf{{S}} = \begin{bmatrix} 0, & 0, & 0, & 1, & -1 \end{bmatrix}$. This restriction may be a known fact from the theory or experiment view.

Direct calculations will show that

and

when we consider above matrices for the model ${\mathcal{L}}$ in (4.5). Now it is easy to see that

i.e.,

Using the numpy library in Python and setting the above findings in the matrix $\mathbf{{E}}_{l}$ in (4.6), we find the eigenvalues of matrix $\mathbf{{E}}_{l}$ as 1.44, 20.98, 20.98, 80.99, -80.99, -20.98, -19.36 and -1.25. Thus, we can see that the number of positive eigenvalues of $\mathbf{{E}}_{l}$ is 4 and the number of negative eigenvalues of $\mathbf{{E}}_{l}$ is 4, i.e., $\boldsymbol{i}_{+}(\mathbf{{E}}_{l}) = 4$ and $\boldsymbol{i}_{-}(\mathbf{{E}}_{l}) = 4$. It is easily seen from (1.5) that $\boldsymbol{r}(\mathbf{{E}}_{l}) = 8$. Also, we obtain $\boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_s, & {\boldsymbol{\Lambda}}_{r} \end{bmatrix} = 4$ and $\boldsymbol{r}(\mathbf{{X}}_{s}) = 3$. Therefore, $\boldsymbol{i}_{+}(\mathbf{{E}}_{l}) = \boldsymbol{r}\begin{bmatrix} \mathbf{{X}}_s, & {\boldsymbol{\Lambda}}_{r} \end{bmatrix} = 4$ $\Leftrightarrow$ ${\rm{MSEM}}(\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{BLUE}}}) \preccurlyeq {\rm{MSEM}}(\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{OLSE}}})$ holds. Since $\boldsymbol{r}(\mathbf{{X}}_{s}) = 3$, i.e., $\boldsymbol{i}_{-}(\mathbf{{E}}_{l})\ne \boldsymbol{r}(\mathbf{{X}}_s)$, ${\rm{MSEM}}(\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{OLSE}}}) \preccurlyeq {\rm{MSEM}}(\widehat{ \mathbf{{X}}_{s} {\boldsymbol{\alpha}}}_{ {\rm{BLUE}}})$ is not provided.

5.   Conclusions

The findings of this study, according to the ${\rm{MSEM}}$ criterion, provide a broad perspective on the ${\rm{SRLMM}}$s in the context of predictor comparability problems using the methodology of block matrix inertias and ranks. We establish the comparison results after converting an explicitly ${\rm{SRLMM}}$ to an implicitly linear stochastically restricted ${\rm{LMM}}$. We also reduce our results to models ${\rm{CLMM}}$s and ${\rm{ULMM}}$s.

We derive the formulas of comparisons of any predictor and the ${\rm{BLUP}}$ for the same general vector of all unknown vectors under ${\rm{SRLMM}}$ by considering the ${\rm{MSEM}}$ sense. Thus, the performance of a new biased or unbiased predictor according to the dispersion matrices of ${\rm{BLUP}}$ can be examined with various inequalities and equalities derived among ${\rm{MSEM}}$s based on inertias and ranks. These kinds of inequalities or equalities for comparisons have a useful and strong statistical explanation in the theory of linear statistical models and applications. The comparisons of MSEMs to determine the effectiveness of any predictor according to BLUP are explained by the comparisons of some quantities that come from inertias and ranks.

As a result, we can say that our contribution to issues involving inference and prediction under an ${\rm{LMM}}$ with a linear stochastic restriction on unknown parameters in the literature is to serve as a theoretically comprehensive and significant search for the comparison results between any predictors and ${\rm{BLUP}}$s.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to express their sincere thanks to the handling editor and anonymous reviewers for their helpful comments and suggestions.

Conflict of interest

The authors state that they have no competing interest.