Further research on simple projection prediction under a general linear model

Abstract

Linear model is a kind of important model in statistics. The estimation and prediction of unknown parameters in the model is a basic problem in statistical inference. According to different evaluation criteria, different forms of predictors are obtained. The simple projection prediction (SPP) boasts simple form while the best linear unbiased prediction (BLUP) enjoys very excellent properties. Naturally, the relationship between the two predictors are considered. We give the necessary and sufficient conditions for the equality of SPP and BLUP. Finally, we study the robustness of SPP and BLUP with respect to covariance matrix and illustrate the application of equivalence conditions by use of error uniform correlation models.

Keywords

Linear model / SPP / BLUP / equivalence / robustness

1 Introduction

Linear model is very important in statistics and have important applications in many fields, such as agriculture, medicine, economics, education, and many others. The estimation and prediction of unknown parameters in models are a fundamental problem in statistical inference. Based on different optimal criteria, statisticians can define different estimators and predictors of unknown parameters. Among these estimators and predictors, classic ones include least squares estimation, weighted least squares estimation, simple projection prediction (SPP), optimal linear unbiased prediction (BLUP), etc. Among them, simple projection prediction boasts a simple form while optimal linear unbiased prediction enjoys excellent properties. Naturally, we consider the topic of equality between them, and present the equivalent conditions for equality, to provide a theoretical basis for further statistical inference.

We consider the following classic linear model

(1.1){\mathcal{M}}_1:\mathbf{y}\mathbf{=}\mathbf{X}\mathit{\beta }\mathbf{+}\mathit{\epsilon }\mathbf{,}\mathrm{E}\left(\mathit{\epsilon }\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathrm{D}\left(\mathit{\epsilon }\right)\mathbf{=}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{.}

Here, \mathbf{y} is an observable random vector of order n\times 1, \mathbf{X} is a model matrix of order n×p under random rank, \mathit{\beta } is an unknown parameter vector of order p\times 1, \mathit{\epsilon } is an error vector of order n\times 1, \mathrm{E}\left(\mathit{\epsilon }\right)=\mathbf{0}\in {\mathbb{R}}^{n\times 1} is a zero mean vector, {\mathbf{\Sigma }}_{11}\in {\mathbb{R}}^{n\times n} is a known non-negative definite matrix, denoted by {\mathbf{\Sigma }}_{11}\ge \mathbf{0}, where E(·) and D(·) represent the expectation and variance of the random vector, respectively.

In statistical research, it is sometimes necessary to study the problem of predictor of censored data in future observations. Let the future observation vector model of model (1.1) be

(1.2){\mathcal{M}}_2:{\mathbf{y}}_{\ast }={\mathbf{X}}_{\ast }\mathit{\beta }+{\mathit{\epsilon }}_{\ast },\mathrm{E}\left({\mathit{\epsilon }}_{\ast }\right)=\mathbf{0},\mathrm{D}\left({\mathit{\epsilon }}_{\ast }\right)={\mathbf{\Sigma }}_{22}.

Here, {\mathbf{y}}_{\ast } is the future observation vector of order m×1, {\mathbf{X}}_{\ast } is the model matrix of order m\times p under random rank, {\mathit{\epsilon }}_{\ast } is the error vector of order m×1, \mathrm{E}\left({\mathit{\epsilon }}_{\ast }\right)=\mathbf{0}\in {\mathbb{R}}^{m\times 1} is the zero mean vector, and {\mathbf{\Sigma }}_{22}\in {\mathbb{R}}^{m\times m} is a known non-negative definite matrix.

The simultaneous model of models (1.1) and (1.2) is

(1.3)\mathcal{M}:\hat{\mathbf{y}}=\mathbf{Z}\mathit{\beta }\mathbf{+}\hat{\mathit{\epsilon }}\mathbf{,}\mathrm{E}\left(\hat{\mathit{\epsilon }}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathrm{D}\left(\hat{\mathit{\epsilon }}\right)\mathbf{=}\left[\begin{array}{cc}{\mathbf{\Sigma }}_{11}& {\mathbf{\Sigma }}_{12}\\ {\mathbf{\Sigma }}_{21}& {\mathbf{\Sigma }}_{22}\end{array}\right]\triangleq \mathrm{V}\mathbf{,}

where \hat{\mathbf{y}}=\left[\genfrac{}{}{0pt}{}{\mathbf{y}}{{\mathbf{y}}_{\ast }}\right], \mathbf{Z}\mathbf{=}\left[\genfrac{}{}{0pt}{}{\mathbf{X}}{{\mathbf{X}}_{\mathbf{\ast }}}\right], \hat{\mathit{\epsilon }}=\left[\genfrac{}{}{0pt}{}{\mathit{\epsilon }}{{\mathit{\epsilon }}_{\ast }}\right].

Due to the presence of unknown parameter vectors and error terms in linear models, people usually only focus on the estimation of unknown parameter \mathit{\beta }, and there have been scarce discussions on the joint prediction of unknown parameter vectors and error terms. Here, we consider the joint prediction problem of unknown parameter vectors and error terms

(1.4)\mathit{\varphi }=\mathbf{F}\mathit{\beta }\mathbf{+}\mathbf{G}\mathit{\epsilon }\mathbf{+}\mathbf{H}{\mathit{\epsilon }}_{\mathbf{\ast }}\mathbf{.}

Here, \mathbf{F}\in {\mathbb{R}}^{\mathbf{s}\times \mathbf{p}}, \mathbf{G}\in {\mathbb{R}}^{\mathbf{s}\times \mathbf{n}} and \mathbf{H}\in {\mathbb{R}}^{\mathbf{s}\times \mathbf{m}} are known matrices. Since \mathbf{F}, \mathbf{G}, and \mathbf{H} can be any matrix, \mathit{\varphi } contains all linear combinations of unknown parameters \mathit{\beta }, error terms \mathit{\epsilon } and {\mathit{\epsilon }}_{\ast }.

(i) If \mathbf{F}\mathbf{=}\mathbf{Z}, \mathbf{G}\mathbf{=}\left[\genfrac{}{}{0pt}{}{{\mathbf{I}}_{\mathbf{n}}}{\mathbf{0}}\right], \mathbf{H}\mathbf{=}\left[\genfrac{}{}{0pt}{}{\mathbf{0}}{{\mathbf{I}}_{\mathbf{m}}}\right], then \mathit{\varphi } is the joint observation vector \hat{\mathbf{y}} in model (1.3);

(ii) If \mathbf{F}\mathbf{=}\mathbf{Q}\mathbf{Z}, \mathbf{G}\mathbf{=}\mathbf{Q}\left[\genfrac{}{}{0pt}{}{{\mathbf{I}}_{\mathbf{n}}}{\mathbf{0}}\right], \mathbf{H}\mathbf{=}\mathbf{Q}\left[\genfrac{}{}{0pt}{}{\mathbf{0}}{{\mathbf{I}}_{\mathbf{m}}}\right], then \mathit{\varphi } is random linear function of the joint vector in model (1.3), where \mathbf{Q}\in {\mathbb{R}}^{\mathbf{t}\times \mathbf{(}\mathbf{m}\mathbf{+}\mathbf{n}\mathbf{)}} is the known matrix;

(iii) If \mathbf{F}\mathbf{=}{\mathbf{1}}_{\mathbf{m}\mathbf{+}\mathbf{n}}^{\prime }\mathbf{Z}, \mathbf{G}\mathbf{=}{\mathbf{1}}_{\mathbf{n}}^{\prime }, \mathbf{H}\mathbf{=}{\mathbf{1}}_{\mathbf{m}}^{\prime }, then \mathit{\varphi } is the sum of the observed values of the joint vector \hat{\mathbf{y}} in model (1.3), i.e., \mathit{\varphi }=\sum _i{\mathbf{y}}_i, where {\mathbf{1}}_k^{\prime } represents vectors of order 1\times k with all components being 1;

(iv) If \mathbf{F}\mathbf{=}{\mathbf{X}}_{\mathbf{\ast }}, \mathbf{G}\mathbf{=}\mathbf{0}, \mathbf{H}\mathbf{=}{\mathbf{I}}_{\mathbf{m}}, then \mathit{\varphi } is the future observation vector {\mathbf{y}}_{\ast } in model (1.2);

(v) If \mathbf{F}\mathbf{=}\mathbf{X}, \mathbf{G}\mathbf{=}{\mathbf{I}}_{\mathbf{n}}, \mathbf{H}\mathbf{=}\mathbf{0}, then \mathit{\varphi } is the known observation vector \mathbf{y} in model (1.1).

Reference [17] gave an important lemma for optimizing matrix-valued functions in the sense of Löwner partial order, and studied the joint prediction problem under mixed effect model. In recent years, references [19, 20] presented the BLUP analytical expressions for the joint prediction of fixed effect, random effect, and error term under random effect model by solving the constrained quadratic matrix-valued function optimization problem in the sense of Löwner partial order. This work can be referred to in references [3-5, 7-11].

With (1.3) and (1.4), we can get

\mathrm{E}\left(\mathit{\varphi }\right)=\mathbf{F}\mathit{\beta }\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\left(\mathit{\varphi }\right)\mathbf{=}\mathbf{J}\mathbf{V}{\mathbf{J}}^{\prime }\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\mathbf{(}\mathit{\varphi }\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{J}\mathbf{V}{\mathbf{T}}^{\prime }\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\left(\mathbf{y}\right)\mathbf{=}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{=}{\mathbf{T}\mathbf{V}\mathbf{T}}^{\prime }\mathbf{,}

where \mathbf{J}\mathbf{=}\mathbf{[}\mathbf{G}\mathbf{,}\mathbf{H}\mathbf{]}, \mathbf{T}\mathbf{=}\mathbf{[}{\mathbf{I}}_{\mathbf{n}}\mathbf{,}\mathbf{0}\mathbf{]}.

This paper uses a novel method to solve the constrained quadratic matrix-valued function optimization problem, establishes a complete theoretical system for optimal linear unbiased prediction (BLUP) with respect to \mathit{\varphi }, and presents an analytical expression for BLUP and related algebraic and statistical properties. The main work is divided into the following four parts:

(I) Provide model (1.1) consistency definition and definition of \mathit{\varphi } and \mathbf{Q}\hat{\mathbf{y}} predictability under model (1.1);

(II) Establish analytical expressions for {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathit{\varphi }\right), {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)and their algebraic and statistical properties;

(III) Present the equivalence conditions for {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right);

(IV) Study the robustness of covariance matrices with respect to \mathrm{S}\mathrm{P}\mathrm{P}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and \mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}\left(\mathbf{Q}\hat{\mathbf{y}}\right).

2 Preliminaries

In this paper, {\mathbf{A}}^{\prime }, r\left(\mathbf{A}\right) and \mathcal{R}\left(\mathbf{A}\right) are used to indicate the transpose, rank of \mathbf{A}\in {\mathbb{R}}^{\mathbf{m}\times \mathbf{n}} matrix, and linear subspace generated by column vector of matrix A, respectively. I_m represents the identity matrix of order m. The Moore-Penrose inverse of matrix A, denoted by A^†, is defined as the unique solution G that simultaneously satisfies the following four matrix equations \mathbf{A}\mathbf{G}\mathbf{A}\mathbf{=}\mathbf{A}, \mathbf{G}\mathbf{A}\mathbf{G}\mathbf{=}\mathbf{G}, (\mathbf{A}\mathbf{G}{\mathbf{)}}^{\prime }\mathbf{=}\mathbf{A}\mathbf{G} and (\mathbf{G}\mathbf{A}{\mathbf{)}}^{\prime }\mathbf{=}\mathbf{G}\mathbf{A}. P_A, E_A, and P_A represent orthogonal projection matrices (symmetric idempotent matrices), and {\mathbf{P}}_{\mathbf{A}}=\mathbf{A}{\mathbf{A}}^{\mathbf{†}}, {\mathbf{E}}_{\mathbf{A}}={\mathbf{A}}^{\perp }={\mathbf{I}}_{\mathbf{m}}-\mathbf{A}{\mathbf{A}}^{\mathbf{†}} and {\mathbf{F}}_{\mathbf{A}}={\mathbf{I}}_{\mathrm{n}}-{\mathbf{A}}^{†}\mathbf{A}. Given two symmetric matrices A and B of the same order, if \mathbf{A}\mathbf{-}\mathbf{B}\succcurlyeq \mathbf{0}, they satisfy Löwner partial order \mathbf{A}\succcurlyeq \mathbf{B}.

Matrix theory, as a mathematical operation tool, has been widely used in linear statistical models (see monograph [14]). The rank of a matrix is a rather basic concept and an important algebraic tool in matrix algebra, which can be used to establish and simplify various equations containing generalized inverse matrix operations. In recent decades, researchers at home and abroad have established numerous analytical calculation formulas for matrix operation ranks, used these formulas to handle the construction, calculation, and comparison of a large number of estimators in regression analysis, and derived many simple and practical criteria for determining the equivalence of different estimators, laying an important theoretical basis for subsequent research on statistical inference of more complex statistical models. Some basic formulas for block matrix rank were used in some cases, such as references [12, 14].

Below are some basic formulas for calculating the rank of block matrices. They will be used in the proof of the theorem later.

Lemma 2.1　 Assume A is a random matrix, then \mathbf{A}\mathbf{=}\mathbf{0}\iff \mathbf{r}\left(\mathbf{A}\right)\mathbf{=}\mathbf{0}.

Lemma 2.2 [12]　 Let \mathbf{A}\in {\mathbb{R}}^{\mathbf{m}\times \mathbf{n}}, \mathbf{B}\in {\mathbb{R}}^{\mathbf{m}\times \mathbf{k}}, \mathbf{C}\in {\mathbb{R}}^{\mathbf{l}\times \mathbf{n}}, \mathbf{D}\in {\mathbb{R}}^{\mathbf{l}\times \mathbf{k}}. Then

(2.1)r[\mathbf{A}\mathbf{,}\mathbf{B}\mathbf{]}\mathbf{=}\mathbf{r}\left(\mathbf{A}\right)\mathbf{+}\mathbf{r}\left({\mathbf{E}}_{\mathbf{A}}\mathbf{B}\right)\mathbf{=}\mathbf{r}\left(\mathbf{B}\right)\mathbf{+}\mathbf{r}\left({\mathbf{E}}_{\mathbf{B}}\mathbf{A}\right)\mathbf{,}

(2.2)r\left[\begin{array}{c}\mathbf{A}\\ \mathbf{C}\end{array}\right]=r\left(\mathbf{A}\right)+r\left(\mathbf{C}{\mathbf{F}}_{\mathbf{A}}\right)=r\left(\mathbf{C}\right)+r\left(\mathbf{A}{\mathbf{F}}_{\mathbf{C}}\right),

(2.3)r\left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\\ \mathbf{C}& \mathbf{D}\end{array}\right]=r\left(\mathbf{A}\right)+r(\mathbf{D}\mathbf{-}\mathbf{C}{\mathbf{A}}^{\mathbf{†}}\mathbf{B}\mathbf{)}\mathbf{,}\mathbf{i}\mathbf{f}\mathcal{R}\left(\mathbf{B}\right)\subseteq \mathcal{R}\left(\mathbf{A}\right)\mathbf{,}\mathcal{R}\left({\mathbf{C}}^{\prime }\right)\subseteq \mathcal{R}\left({\mathbf{A}}^{\prime }\right)\mathbf{.}

And the following results hold:

(a) r[\mathbf{A}\mathbf{,}\mathbf{B}\mathbf{]}\mathbf{=}\mathbf{r}\left(\mathbf{A}\right)\iff \mathcal{R}\left(\mathbf{B}\right)\subseteq \mathcal{R}\left(\mathbf{A}\right)\iff \mathbf{A}{\mathbf{A}}^{\mathbf{†}}\mathbf{B}\mathbf{=}\mathbf{B}\iff {\mathbf{E}}_{\mathbf{A}}\mathbf{B}\mathbf{=}\mathbf{0}\mathbf{.}

(b) r\left[\mathbf{A}\right]=r\left(\mathbf{A}\right)\iff \mathcal{R}\left({\mathbf{C}}^{\prime }\right)\subseteq \mathcal{R}\left({\mathbf{A}}^{\prime }\right)\iff {\mathbf{C}\mathbf{A}}^{†}\mathbf{A}\mathbf{=}\mathbf{C}\iff {\mathbf{C}\mathbf{F}}_{\mathbf{A}}\mathbf{=}\mathbf{0}\mathbf{.}

Lemma 2.3 [22]　 Let \mathbf{A}\in {\mathbb{R}}^{\mathbf{m}\times \mathbf{n}}, \mathbf{B}\in {\mathbb{R}}^{\mathbf{m}\times \mathbf{k}}, and assume \mathcal{R}\left(\mathbf{A}\right)=\mathcal{R}\left(\mathbf{B}\right). Then

\mathbf{X}\mathbf{A}\mathbf{=}\mathbf{0}\iff \mathbf{X}\mathbf{B}\mathbf{=}\mathbf{0}\mathbf{.}

Lemma 2.4 [13]　 The necessary and sufficient condition for the linear matrix equation \mathbf{A}\mathbf{X}\mathbf{=}\mathbf{B} to be solvable is r[\mathbf{A}\mathbf{,}\mathbf{B}\mathbf{]}\mathbf{=}\mathbf{r}\left(\mathbf{A}\right), or \mathbf{A}{\mathbf{A}}^{\mathbf{†}}\mathbf{B}\mathbf{=}\mathbf{B}. Then the general solution of the equation can be expressed as \mathbf{X}\mathbf{=}{\mathbf{A}}^{\mathbf{†}}\mathbf{B}\mathbf{+}\mathbf{(}\mathbf{I}\mathbf{-}{\mathbf{A}}^{\mathbf{†}}\mathbf{A}\mathbf{)}\mathbf{U}, where U is an arbitrary matrix.

Below is the definition of model (1.1) consistency. With Moore-Penrose inverse definition, equation

[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{]}}^{\mathbf{†}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}\mathbf{=}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}

always holds. Thus,

\begin{array}{rl}& \mathrm{E}\left(\right({\mathbf{I}}_n-[\mathbf{X},{\mathbf{\Sigma }}_{11}][\mathbf{X},{\mathbf{\Sigma }}_{11}{]}^{†})\mathbf{y})=({\mathbf{I}}_n-[\mathbf{X},{\mathbf{\Sigma }}_{11}][\mathbf{X},{\mathbf{\Sigma }}_{11}{]}^{†})X\mathit{\beta }=\mathbf{0}\mathbf{,}\\ & \mathrm{C}\mathrm{o}\mathrm{v}\left(\right({\mathbf{I}}_n-[\mathbf{X},{\mathbf{\Sigma }}_{11}][\mathbf{X},{\mathbf{\Sigma }}_{11}{]}^{†})\mathbf{y})\\ & =({\mathbf{I}}_n-[\mathbf{X},{\mathbf{\Sigma }}_{11}][\mathbf{X},{\mathbf{\Sigma }}_{11}{]}^{†}){\mathbf{\Sigma }}_{11}({\mathbf{I}}_n-[\mathbf{X},{\mathbf{\Sigma }}_{11}\left]\right[\mathbf{X},{\mathbf{\Sigma }}_{11}{]}^{†}{)}^{\prime }=0\end{array}

is established. These two equations indicate that [\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{]}}^{\mathbf{†}}\mathbf{y}\mathbf{=}\mathbf{y} holds true for a probability of 1, or \mathbf{y}\in \mathcal{R}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]} holds true for a probability of 1. For the definition of model consistency, please refer to [15, 16].

Definition 2.1　If \mathbf{y}\in \mathcal{R}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]} holds true for a probability of 1, then model {\mathcal{M}}_1 is consistent or compatible.

Below is a classic definition of predictability (see [1, 6]).

Definition 2.2　If there exists a linear estimator Ly that satisfies

\mathrm{E}(\mathbf{L}\mathbf{y}\mathbf{-}\mathit{\varphi }\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,}

then vector \mathit{\varphi } is linearly predictable under model {\mathcal{M}}_1.

Theorem 2.1　 \mathit{\varphi } is linearly estimable under model {\mathcal{M}}_1\iff \mathbf{F}\mathit{\beta } is linearly estimable under model {\mathcal{M}}_1\iff \mathcal{R}\left({\mathbf{F}}^{\prime }\right)\subseteq \mathcal{R}\left({\mathbf{X}}^{\prime }\right). In particular,

(a) y is always linearly predictable under model, and \mathbf{X}\mathit{\beta } is always linearly estimable under model {\mathcal{M}}_1;

(b) \hat{\mathbf{y}} and {\mathbf{y}}_{\ast } are linearly predictable under model {\mathcal{M}}_1\iff \mathbf{Z}\mathit{\beta }\mathbf{,}{\mathbf{X}}_{\mathbf{\ast }}\mathit{\beta } is linearly estimable under model {\mathcal{M}}_1\iff \mathcal{R}\left({\mathbf{X}}_{\ast }^{\prime }\right)\subseteq \mathcal{R}\left({\mathbf{X}}^{\prime }\right);

(c) \mathbf{G}\hat{\mathbf{y}} is linearly predictable under model {\mathcal{M}}_1\iff \mathbf{G}\mathbf{Z}\mathit{\beta } is linearly estimable under model {\mathcal{M}}_1\iff \mathcal{R}\left(\right(\mathbf{G}\mathbf{Z}{\mathbf{)}}^{\prime }\mathbf{)}\subseteq \mathcal{R}\mathbf{(}{\mathbf{X}}^{\prime }\mathbf{)}.

Proof　Through Definition 2.2, \mathit{\Phi } is linearly predictable under model {\mathcal{M}}_1\iff \mathbf{F}\mathit{\beta } is linearly estimable under model {\mathcal{M}}_1\iff \mathbf{L}\mathbf{X}\mathit{\beta }\mathbf{=}\mathbf{F}\mathit{\beta }\iff \mathbf{L}\mathbf{X}\mathbf{=}\mathbf{F}.

With Lemma 2.4, the result holds. □

Many topics in statistics are closely related to optimization problems in mathematics. References [19‒21] provided an analytical solution for the optimization problem of quadratic matrix-valued functions with constraints, and used the following lemma to present an analytical expression for joint prediction of BLUP under the most general assumptions of the model, as well as algebraic and statistical properties.

Lemma 2.5　 Let \mathbf{A}\in {\mathbb{R}}^{\mathbf{p}\times \mathbf{q}}, \mathbf{B}\in {\mathbb{R}}^{\mathbf{n}\times \mathbf{q}}, \mathbf{C}\in {\mathbb{R}}^{\mathbf{p}\times \mathbf{m}}, \mathbf{D}\in {\mathbb{R}}^{\mathbf{n}\times \mathbf{m}}, \mathbf{M}\in {\mathbb{R}}^{\mathbf{m}\times \mathbf{m}} be non-negative definite matrices. Assume matrix equation \mathbf{G}\mathbf{A}\mathbf{=}\mathbf{B} is solvable with respect to \mathbf{G}\in {\mathbb{R}}^{\mathbf{n}\times \mathbf{p}}, then in the sense of Löwner partial order, the quadratic matrix-valued function with constraint

(2.4)f\left(\mathbf{G}\right)=(\mathbf{G}\mathbf{C}\mathbf{+}\mathbf{D}\mathbf{)}\mathbf{M}\mathbf{(}\mathbf{G}\mathbf{C}\mathbf{+}\mathbf{D}{\mathbf{)}}^{\prime }⪰\mathbf{f}\left({\mathbf{G}}_{\mathbf{0}}\right)\mathrm{s}\mathbf{.}\mathrm{t}\mathbf{.}\mathbf{G}\mathbf{A}\mathbf{=}\mathbf{B}

is always solvable with respect to G, and the solution G₀ of (2.4) satisfies the following matrix equation

{\mathbf{G}}_0[\mathbf{A}\mathbf{,}{\mathbf{C}\mathbf{M}\mathbf{C}}^{\prime }{\mathbf{A}}^{\perp }\mathbf{]}\mathbf{=}\mathbf{[}\mathbf{B}\mathbf{,}\mathbf{-}{\mathbf{D}\mathbf{M}\mathbf{C}}^{\prime }{\mathbf{A}}^{\perp }\mathbf{]}\mathbf{.}

Then, the analytical expression of G₀, f(G₀), and f(G) can be expressed as

{\mathbf{G}}_0=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathbf{G}\mathbf{A}\mathbf{=}\mathbf{B}}{\mathrm{m}\mathrm{i}\mathrm{n}} f\left(\mathbf{G}\right)=[\mathbf{B}\mathbf{,}\mathbf{-}{\mathbf{D}\mathbf{M}\mathbf{C}}^{\prime }{\mathbf{A}}^{\perp }\mathbf{]}\mathbf{[}\mathbf{A}\mathbf{,}{\mathbf{C}\mathbf{M}\mathbf{C}}^{\prime }{\mathrm{A}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}\mathbf{U}\mathbf{[}\mathbf{A}\mathbf{,}{\mathbf{C}\mathbf{M}\mathbf{C}}^{\prime }{\mathbf{]}}^{\perp }\mathbf{,}

f\left({\mathbf{G}}_0\right)=\underset{\mathbf{G}\mathbf{A}\mathbf{=}\mathbf{B}}{\mathrm{m}\mathrm{i}\mathrm{n}} f\left(\mathbf{G}\right)={\mathbf{W}\mathbf{M}\mathbf{W}}^{\prime }-{\mathbf{W}\mathbf{M}\mathbf{C}}^{\prime }({\mathbf{A}}^{\perp }{\mathbf{C}\mathbf{M}\mathbf{C}}^{\prime }{\mathbf{A}}^{\perp }{)}^{†}{\mathbf{C}\mathbf{M}\mathbf{W}}^{\prime },

f\left(\mathbf{G}\right)=f\left({\mathbf{G}}_0\right)+(\mathbf{G}\mathbf{C}\mathbf{+}\mathbf{D}\mathbf{)}\mathbf{M}{\mathbf{C}}^{\prime }\left({\mathbf{A}}^{\perp }\mathbf{C}\mathbf{M}{\mathbf{C}}^{\prime }{\mathbf{A}}^{\perp }{\mathbf{)}}^{\mathbf{†}}\mathbf{C}\mathbf{M}\right(\mathbf{G}\mathbf{C}\mathbf{+}\mathbf{D}{\mathbf{)}}^{\prime }\mathbf{,}

where \mathbf{W}\mathbf{=}{\mathbf{B}\mathbf{A}}^{\mathbf{†}}\mathbf{C}\mathbf{+}\mathbf{D}\mathbf{,}\mathbf{U}\in {\mathbb{R}}^{\mathbf{n}\times \mathbf{p}} is a random matrix.

3 Predictable vector \mathit{\phi } under model {\mathcal{M}}_1 and its special cases SPP and BLUP

Prediction in linear models has always been a fundamental problem in statistics. Based on different evaluation criteria, different expression forms of predictor can be obtained. The intuitive idea is to seek a concise form of prediction. Särndal and Wright [18] emphasized that prediction should have a concise and intuitive form, and the most concise and intuitive prediction was SPP; another commonly used optimal criterion was to find the prediction with the smallest covariance among all unbiased predictors of unknown parameters in the sense of Löwner partial order, which was the famous BLUP. This paper mainly considers the topic of equality between {\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right). The work in this regard can be seen in [2, 18, 23-25].

Firstly, we provide the analytical expressions for these two basic predictors.

Definition 3.1　If there exists a matrix \hat{\mathbf{L}} that under Löwner partial order satisfies

\mathrm{C}\mathrm{o}\mathrm{v}\left(\mathbf{L}\mathbf{y}\mathbf{-}\mathit{\varphi }\right)⪰\mathrm{C}\mathrm{o}\mathrm{v}\left(\hat{\mathbf{L}}\mathbf{y}\mathbf{-}\mathit{\varphi }\right)\mathrm{s}.\mathrm{t}.\mathrm{E}(\mathbf{L}\mathbf{y}\mathbf{-}\mathit{\varphi }\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,}

then linear statistic \hat{\mathbf{L}}\mathbf{y} is defined as BLUP of \mathit{\varphi }, denoted by

\hat{\mathbf{L}}\mathbf{y}\mathbf{=}{\mathbf{\text{BLUP}}}_{{\mathcal{M}}_{\mathbf{1}}}\left(\mathit{\varphi }\right)\mathbf{=}{\mathbf{\text{BLUP}}}_{{\mathcal{M}}_{\mathbf{1}}}\mathbf{(}\mathbf{F}\mathit{\beta }\mathbf{+}\mathbf{G}\mathit{\epsilon }\mathbf{+}\mathbf{H}{\mathit{\epsilon }}_{\mathbf{\ast }}\mathbf{)}\mathbf{.}

When \mathbf{G}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{H}\mathbf{=}\mathbf{0}\mathbf{,}\hat{\mathbf{L}} is defined as BLUE of \mathbf{F}\mathit{\beta }, denoted by

\hat{\mathbf{L}}\mathbf{y}\mathbf{=}{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_{\mathbf{1}}}\left(\mathbf{F}\mathit{\beta }\right)\mathbf{.}

When F=QZ, \mathbf{G}\mathbf{=}\mathbf{Q}\left[\genfrac{}{}{0pt}{}{{\mathbf{I}}_{\mathbf{n}}}{\mathbf{0}}\right], \mathbf{H}\mathbf{=}\mathbf{Q}\left[\genfrac{}{}{0pt}{}{\mathbf{0}}{{\mathbf{I}}_{\mathbf{m}}}\right], \hat{\mathbf{L}} is defined as BLUP of Q\hat{\mathbf{y}}, denoted by

\hat{\mathbf{L}}\mathbf{y}\mathbf{=}{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_{\mathbf{1}}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)\mathbf{.}

The derivation of the analytical expressions for BLUP/BLUE under model {\mathcal{M}}_1 with respect to \mathit{\varphi } and its special cases can be found in [19].

Lemma 3.1　 Assume \mathit{\varphi } is predictable under model {\mathcal{M}}_1, that is, \mathcal{R}\left({\mathbf{F}}^{\prime }\right)\subseteq \mathcal{R}\left({\mathbf{X}}^{\prime }\right), then matrix equation

\mathbf{L}\mathbf{[}\mathbf{X}\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\mathbf{(}\mathbf{y}\left){\mathbf{X}}^{\perp }\right]\mathbf{=}\mathbf{[}\mathbf{F}\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\{\mathit{\varphi }\mathbf{,}\mathbf{y}\left\}{\mathbf{X}}^{\perp }\right]

is solvable, and the general solution of the equation and the BLUP analytical expression of \mathit{\varphi } are

{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathit{\varphi }\right)=\mathbf{L}\mathbf{y}\mathbf{=}\left(\mathbf{[}\mathbf{F}\mathbf{,}{\mathbf{C}\mathbf{X}}^{\perp }\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}\mathbf{U}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\right)\mathbf{y}\mathbf{,}

where \mathbf{C}\mathbf{=}\mathrm{C}\mathrm{o}\mathrm{v}\{\mathit{\varphi }\mathbf{,}\mathbf{y}\}, \mathrm{C}\mathrm{o}\mathrm{v}\left(\mathbf{y}\right)={\mathbf{\Sigma }}_{11}, while \mathbf{U}\in {\mathbb{R}}^{\mathbf{s}\times \mathbf{n}} is a random matrix.

Assume Q\hat{\mathbf{y}} is predictable under model {\mathcal{M}}_1, QZβ is estimable under model {\mathcal{M}}_1, that is, \mathcal{R}\left(\right(\mathbf{Q}\mathbf{Z}{\mathbf{)}}^{\prime }\mathbf{)}\subseteq \mathcal{R}\mathbf{(}{\mathbf{X}}^{\prime }\mathbf{)}, then matrix equation

\mathbf{L}\mathbf{[}\mathbf{X}\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\mathbf{(}\mathbf{y}\left){\mathbf{X}}^{\perp }\right]\mathbf{=}\mathbf{[}\mathbf{Q}\mathbf{Z}\mathbf{,}\mathrm{C}\mathrm{o}\mathrm{v}\{\mathbf{Q}\hat{\mathbf{y}}\mathbf{,}\mathbf{y}\left\}{\mathbf{X}}^{\perp }\right]

is solvable, so that the BLUP/BLUE analytical expressions of \mathbf{Q}\hat{\mathbf{y}}, \mathbf{Q}\mathbf{Z}\mathit{\beta }, and Q\hat{\mathit{\epsilon }} are

\begin{array}{rl}& {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)=\left([\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}{\mathbf{U}}_{\mathbf{1}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\right)\mathbf{y}\mathbf{,}\\ & {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\mathbf{Z}\mathit{\beta }\right)=\left([\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}{\mathbf{U}}_{\mathbf{2}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\right)\mathbf{y}\mathbf{,}\\ & {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right)=\left(\begin{array}{c}[\mathbf{0},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][X,{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_3[X,{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\end{array}\right)\mathbf{y}\mathbf{,}\end{array}

where \mathbf{T}\mathbf{=}\mathbf{[}{\mathbf{I}}_{\mathbf{n}}\mathbf{,}\mathbf{0}\mathbf{]}, \mathrm{C}\mathrm{o}\mathrm{v}\{\mathbf{Q}\hat{\mathbf{y}}\mathbf{,}\mathbf{y}\}\mathbf{=}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }, while {\mathbf{U}}_i\in {\mathbb{R}}^{t\times n} is a random matrix. And the following results are established:

(a) \mathcal{R}[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\mathbf{=}\mathcal{R}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{X}}^{\perp }{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}\mathbf{=}\mathcal{R}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}\mathbf{.}

(b) The only necessary and sufficient condition of coefficient matrix L is r[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\mathbf{=}\mathbf{n}.

(c) The only necessary and sufficient condition \mathbf{y}\in \mathcal{R}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]} of {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathit{\varphi }\right) holds true with a probability of 1, namely model {\mathcal{M}}_1 is compatible.

(d) The following formula holds:

\begin{array}{rl}& \mathrm{C}\mathrm{o}\mathrm{v}({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}(\mathit{\varphi }))=[\mathbf{F},|{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}{\mathbf{\Sigma }}_{11}([\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}{)}^{\prime },\\ & \mathrm{C}\mathrm{o}\mathrm{v}\{{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}(\mathit{\varphi }),\mathit{\varphi }\}=([\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}+\mathbf{U}[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }){\mathbf{C}}^{\prime }\\ & =([\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{T}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }{]}^{†}+\mathbf{U}[X,{\mathbf{T}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }{]}^{\perp }){\mathbf{T}\mathbf{V}\mathbf{J}}^{\prime }\\ & =([\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{T}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }{]}^{†}+\mathbf{U}[\mathbf{X},\mathbf{T}\mathbf{V}{]}^{\perp }){\mathbf{T}\mathbf{V}\mathbf{J}}^{\prime }\\ & =[\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{T}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }{]}^{†}{\mathbf{T}\mathbf{V}\mathbf{J}}^{\prime }\\ & =[\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}{\mathbf{C}}^{\prime },\end{array}

and

\begin{array}{rl}\mathrm{C}\mathrm{o}\mathrm{v}\left(\mathit{\varphi }\right)& -\mathrm{C}\mathrm{o}\mathrm{v}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\right(\mathit{\varphi }\left)\right)\\ & ={\mathbf{J}\mathbf{V}\mathbf{J}}^{\prime }-[\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}{\mathbf{\Sigma }}_{11}([\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{X}}^{\perp }{]}^{†}{)}^{\prime },\end{array}

\begin{array}{rl}& \mathrm{C}\mathrm{o}\mathrm{v}(\mathit{\varphi }-{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}(\mathit{\varphi }\left)\right)\\ & =\left(\right[\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }\left]\right[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}\mathbf{T}-\mathbf{J}\left)\mathbf{V}\right([\mathbf{F},{\mathbf{C}\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}\mathbf{T}-\mathbf{J}{)}^{\prime },\end{array}

where \mathbf{T}\mathbf{=}\mathbf{[}{\mathbf{I}}_{\mathbf{n}}\mathbf{,}\mathbf{0}\mathbf{]}.

(e) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathit{\varphi }\right) can be decomposed into

\begin{array}{rl}& {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathit{\varphi }\right)={\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\left(\mathbf{F}\mathit{\beta }\right)+{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{G}\mathit{\epsilon }\right)+{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{H}{\mathit{\epsilon }}_{\ast }\right),\\ & \mathrm{C}\mathrm{o}\mathrm{v}\left\{{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\right(\mathbf{F}\mathit{\beta }),{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}(\mathbf{G}\mathit{\epsilon }+\mathbf{H}{\mathit{\epsilon }}_{\ast }\left)\right\}=\mathbf{0},\\ & \mathrm{C}\mathrm{o}\mathrm{v}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\right(\mathit{\varphi }\left)\right)=\mathrm{C}\mathrm{o}\mathrm{v}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\right(\mathbf{F}\mathit{\beta }\left)\right)+\mathrm{C}\mathrm{o}\mathrm{v}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\right(G\mathit{\epsilon }+\mathbf{H}{\mathit{\epsilon }}_{\ast }\left)\right).\end{array}

When the covariance matrix V is a special form of matrix, such as \mathbf{V}\mathbf{=}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathbf{(}{\mathbf{V}}_{\mathbf{1}}\mathbf{,}{\mathbf{V}}_{\mathbf{2}}\mathbf{)} or \mathbf{V}\mathbf{=}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathbf{(}{\sigma }_{\mathbf{1}}^{\mathbf{2}}{\mathbf{I}}_{\mathbf{n}}\mathbf{,}{\sigma }_{\mathbf{2}}^{\mathbf{2}}{\mathbf{I}}_{\mathbf{m}}\mathbf{)}, the above results about BLUP/BLUE can be further simplified. The above results are given under the most general assumptions of the model, which do not require conditions like full rank of the model matrix columns and invertibility of the covariance matrix, and there are no assumptions about the distribution of error terms. Therefore, the above lemma displays generality.

Next, we define the simple projection prediction of \mathbf{Q}\hat{\mathbf{y}} under model (1.3).

Definition 3.2　The SPP of vector \mathbf{Q}\hat{\mathbf{y}} under model (1.3) is defined as

{\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\mathbf{Z}\mathit{\beta }\right)=\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\left]\right[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}\mathbf{U}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{)}\mathbf{y}\mathbf{,}

and

\mathrm{E}\left({\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\right(\mathbf{Q}\hat{\mathbf{y}}\left)\right)\mathbf{=}\mathrm{E}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_{\mathbf{1}}}\right(\mathbf{Q}\mathbf{Z}\mathit{\beta }\left)\right)\mathbf{=}\mathrm{E}\left(\mathbf{Q}\hat{\mathbf{y}}\right)\mathbf{=}\mathbf{Q}\mathbf{Z}\mathit{\beta }\mathbf{.}

4 Research on the relationship between BLUP and SPP of Q\hat{\mathbf{y}}

BLUP and SPP, as two basic predictors in mathematical statistics, are based on different optimal criteria and therefore have different analytical expressions. BLUP enjoys many excellent algebraic and statistical properties, and it has the minimum variance in unbiased predictors. Statisticians often use BLUP to measure the effectiveness of other estimators. In order to study the relationship between the two linear estimators G₁y and G₂y, the following definition is given first.

Definition 4.1　Let y be a random vector, G₁ and G₂ be two matrices with appropriate order.

(a) If {\mathbf{G}}_1={\mathbf{G}}_2, then the two random vectors G₁y and G₂y are absolutely equal, namely {\mathbf{G}}_1\mathbf{y}\mathbf{=}{\mathbf{G}}_{\mathbf{2}}\mathbf{y}.

(b) If \mathrm{E}({\mathbf{G}}_1\mathbf{y}\mathbf{-}{\mathbf{G}}_{\mathbf{2}}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{0} and \mathrm{C}\mathrm{o}\mathrm{v}({\mathbf{G}}_1\mathbf{y}\mathbf{-}{\mathbf{G}}_{\mathbf{2}}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{0} hold simultaneously, then the two random vectors G₁y and G₂y are equal with a probability of 1.

The main conclusions of this paper are presented below.

Theorem 4.1　 Assume Q\hat{\mathbf{y}} is linearly predictable under model {\mathcal{M}}_1, then the following conclusions are equivalent:

(i) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right).

(ii) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right) are equal with a probability of 1.

(iii) \mathrm{D}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\right(\mathit{Q}\hat{\mathit{\epsilon }}\left)\right)=\mathbf{0}.

(iv) {\mathrm{D}(\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right))={\mathrm{D}(\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}(\mathbf{Q}\hat{\mathbf{y}}\left)\right)\mathbf{.}

(v) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right)=\mathbf{0}.

(vi) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right)=\mathbf{0} holds with a probability of 1.

(vii) \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\mathbf{=}\mathbf{0}\mathbf{.}

(viii) r[{\mathbf{T}\mathbf{V}\mathbf{Q}}^{\prime },\mathit{X}]=r\left(\mathbf{X}\right).

(ix) \mathcal{R}({\mathbf{\Sigma }}_{11}{\mathbf{Q}}_1^{\prime }+{\mathbf{\Sigma }}_{12}{\mathbf{Q}}_2^{\prime })\subseteq \mathcal{R}\left(\mathbf{X}\right), where \mathbf{Q}\mathbf{=}\mathbf{[}{\mathbf{Q}}_{\mathbf{1}}\mathbf{,}{\mathbf{Q}}_{\mathbf{2}}\mathbf{]}.

Proof　With Definition 4.1(a), {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right), that is, the coefficient matrices of the two estimators are equal. Hence

(4.1)\begin{array}{rl}& \text{[}\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_1[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & =[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_2[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & \iff {\mathbf{U}}_1[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }-{\mathbf{U}}_2[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & =[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}-[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}\\ & \iff ({\mathbf{U}}_1-{\mathbf{U}}_2)[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }=[\mathbf{0},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}.\end{array}

There exist U₁ and U₂ that satisfy the necessary and sufficient condition of (4.1)

r\left[\begin{array}{c}\left[\mathbf{0}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right]=r\left({\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\right).

After simplification of both sides of (4.2) through elementary transformation of (2.1) and block matrix, we can get

\begin{array}{rl}r\left[\begin{array}{c}\left[\mathbf{0}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{X}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right]& =r\left[\begin{array}{cc}X\left[\mathbf{0}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}& \mathbf{0}\\ {\mathbf{I}}_{\mathbf{n}}& \left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & -r\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{cc}\mathbf{0}& \left[\mathbf{0}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_{\mathbf{n}}& \left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{cc}\mathbf{0}& \left[\mathbf{0}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_{\mathbf{n}}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\right]\\ & =r\left({\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right)+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\right],\end{array}

and

r\left(\right[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{)}\mathbf{=}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{]}\mathbf{.}

Based on the above results, as well as (2.2), Lemma 2.1 and Lemma 2.2(b), we can get

r\left({\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\right)=0\iff r\left[\begin{array}{c}{\mathbf{Q}\mathbf{V}}^{\mathrm{\prime }}{\mathbf{T}}^{\mathrm{\prime }}\\ {\mathbf{X}}^{\mathrm{\prime }}\end{array}\right]=r(\mathbf{X}\mathbf{)}\iff {\mathbf{Q}\mathbf{V}\mathbf{T}}^{\mathrm{\prime }}{\mathbf{X}}^{\perp }\mathbf{=}\mathbf{0}\mathbf{,}

which indicates results (i), (vii), (viii), and (ix) are equivalent.

Through Definition 4.1(b),

{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right) are equal with a probability of 1

\iff [\mathbf{0},{\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }\mathbf{X}}^{\perp }][\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}{\mathbf{\Sigma }}_{\mathbf{11}}\mathbf{=}\mathbf{0}\iff {\mathrm{D}\mathbf{[}\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_{\mathbf{1}}}\mathbf{(}\mathbf{Q}\hat{\mathit{\epsilon }}\left)\right]\mathbf{=}\mathbf{0}\mathbf{.}

From (2.3),

\begin{array}{rl}r\left(\right[\mathbf{0},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }{]}^{†}{\mathbf{\Sigma }}_{11})& =r\lfloor \begin{array}{cc}[\mathbf{X}\mathbf{,}{\mathbf{\Sigma }}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}& {\mathbf{\Sigma }}_{11}\\ [\mathbf{0},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }]& \mathbf{0}\end{array}\rfloor -r[\mathbf{X},{\mathbf{\Sigma }}_{11}{\mathbf{X}}^{\perp }]\\ & =r\left[\begin{array}{ccc}\mathbf{X}& \mathbf{0}& {\mathbf{\Sigma }}_{11}\\ \mathbf{0}& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }& \mathbf{0}\end{array}\right]-r[\mathbf{X},{\mathbf{\Sigma }}_{11}]\\ & =r\left({\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }\mathbf{X}}^{\perp }\right)+r[\mathbf{X},{\mathbf{\Sigma }}_{11}]-r[\mathbf{X},{\mathbf{\Sigma }}_{11}]\\ & =r\left({\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right).\end{array}

Through (2.2), Lemma 2.1 and Lemma 2.2(b), we can obtain

r\left(\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right)=\mathbf{0}\iff r\left[\begin{array}{c}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }\\ {\mathbf{X}}^{\prime }\end{array}\right]=r\left(\mathbf{X}\right)\iff \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\mathbf{=}\mathbf{0}\mathbf{,}

which indicate results (ii), (iii), (vi), (vii), (viii), and (ix) are equivalent. From Lemma 3.1(e),

\begin{array}{rl}& {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\mathbf{Z}\mathit{\beta }\right)+\mathrm{B}\mathrm{L}\left|{\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\right(\mathbf{Q}\hat{\mathit{\epsilon }}\mathbf{)}\mathbf{=}{\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\mathbf{(}\mathbf{Q}\hat{\mathbf{y}}\mathbf{)}\mathbf{+}{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_{\mathbf{1}}}\mathbf{(}\mathbf{Q}\hat{\mathit{\epsilon }}\mathbf{)}\mathbf{,}\\ & {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)\iff {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right)=\mathbf{0},\\ & {\mathrm{D}(\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right))=\mathrm{D}({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{E}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\mathbf{Z}\mathit{\beta }\right))+\mathrm{D}({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right))\\ & =\mathrm{D}\left({\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)\right)+\mathrm{D}\left({\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right)\right),\\ & {\mathrm{D}(\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\mathbf{Q}\hat{\mathit{\epsilon }}\right))=\mathbf{0}\iff {\mathrm{D}(\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}(\mathbf{Q}\hat{\mathbf{y}}\left)\right)\mathbf{=}{\mathrm{D}\mathbf{(}\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)\mathbf{)}\mathbf{,}\end{array}

which indicate results (i), (iii), (iv), and (v) are equivalent. □

From Theorem 4.1, the following corollaries can be obtained.

Corollary 4.1　 Assume \hat{\mathbf{y}} and {\mathbf{1}}_{m+n}^{\prime }\hat{\mathbf{y}} are predictable under model {\mathcal{M}}_1, then the following conclusions hold.

(a) The following conclusions are equivalent:

(i) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left(\hat{\mathbf{y}}\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left(\hat{\mathbf{y}}\right)(with a probability of 1).

(ii) \mathcal{R}\left(\mathbf{T}\mathbf{V}\right)\subseteq \mathcal{R}\left(\mathbf{X}\right).

(b) The following conclusions are equivalent:

(i) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left({\mathbf{y}}_{\ast }\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left({\mathbf{y}}_{\ast }\right)(with a probability of 1).

(ii) \mathcal{R}\left({\mathbf{\Sigma }}_{12}\right)\subseteq \mathcal{R}\left(\mathbf{X}\right).

(c) The following conclusions are equivalent:

(i) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{M}}_1}\left({\mathbf{y}}_{\ast }\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\mathcal{M}}\left({\mathbf{y}}_{\ast }\right)(with a probability of 1).

(ii) {\mathbf{\Sigma }}_{11}{\mathbf{1}}_n+{\mathbf{\Sigma }}_{12}{\mathbf{1}}_m\in \mathcal{R}\left(\mathbf{X}\right).

5 Robustness of SPP and BLUP with respect to covariance matrix

Robustness refers to the relative stability of statistical inference regarding statistical models, namely assumptions. When there is a slight change in model conditions, statistical inference also undergoes a slight change accordingly, indicating the robustness of statistical inference regarding such slight changes. For example, assuming that the variance of the error term in the Gauss-Markov model \mathbf{y}=\mathbf{X}\mathit{\beta }+\mathit{\epsilon } is \mathrm{C}\mathrm{o}\mathrm{v}\left(\mathit{\epsilon }\right)={\sigma }^2{\mathbf{I}}_n, but in practical problems, it is sometimes impossible to fully satisfy this assumption. We can only judge whether \mathrm{C}\mathrm{o}\mathrm{v}\left(\mathit{\epsilon }\right)={\sigma }^2{\mathbf{I}}_n is generally acceptable through analysis or test. We hope that when the difference between the true \mathrm{C}\mathrm{o}\mathrm{v}\left(\mathit{\epsilon }\right) and {\sigma }^2{\mathbf{I}}_n is not too large, the estimator of unknown parameters or their functions will still maintain their original optimality or even if it is not optimal, it will not become too bad, that is, this estimator is robust regarding the covariance matrix. Robustness is a property that every statistical inference should possess. This Section mainly considers the robustness of SPP(Q\hat{\mathbf{y}}) and BLUP(Q\hat{\mathbf{y}}) as well as their special cases regarding covariance matrix. Finally, a specific example is used to illustrate the specific application of the equivalent condition using the error uniform correlation model.

We consider the following two linear models:

(5.1){\mathcal{N}}_i:\mathbf{y}\mathbf{=}\mathbf{X}\mathit{\beta }\mathbf{+}{\mathit{\epsilon }}_{\mathbf{i}}\mathbf{,}\mathrm{E}\left({\mathit{\epsilon }}_{\mathbf{i}}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathrm{D}\left({\mathit{\epsilon }}_{\mathbf{1}}\right)\mathbf{=}{\mathbf{V}}_{\mathbf{11}}\mathbf{,}\mathrm{D}\left({\mathit{\epsilon }}_{\mathbf{2}}\right)\mathbf{=}{\mathbf{W}}_{\mathbf{11}}\mathbf{,}\mathbf{i}\mathbf{=}1,2\mathbf{.}

The future observation vector model of model (5.1) is

(5.2){\mathcal{N}}_{i_{\ast }}:{\mathbf{y}}_{\ast }={\mathbf{X}}_{\ast }\mathit{\beta }+{\mathit{\epsilon }}_{i_{\ast }},\mathrm{E}\left({\mathit{\epsilon }}_{i_{\ast }}\right)=\mathbf{0},\mathrm{D}\left({\mathit{\epsilon }}_{1_{\ast }}\right)={\mathbf{V}}_{22},\mathrm{D}\left({\mathit{\epsilon }}_{2_{\ast }}\right)={\mathbf{W}}_{22},i=1,2.

The simultaneous model of models (5.1) and (5.2) is

\widehat{{\mathcal{N}}_1}:\hat{\mathbf{y}}=\mathbf{Z}\mathit{\beta }\mathbf{+}{\hat{\mathit{\epsilon }}}_{\mathbf{1}}\mathbf{,}\mathrm{E}\left({\hat{\mathit{\epsilon }}}_{\mathbf{1}}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathrm{D}\left({\hat{\mathit{\epsilon }}}_{\mathbf{1}}\right)\mathbf{=}\left[\begin{array}{cc}{\mathbf{V}}_{11}& {\mathbf{V}}_{12}\\ {\mathbf{V}}_{21}& {\mathbf{V}}_{22}\end{array}\right]\triangleq \mathbf{V}\mathbf{,}

\widehat{{\mathcal{N}}_2}:\hat{\mathbf{y}}=\mathbf{Z}\mathit{\beta }\mathbf{+}{\hat{\mathit{\epsilon }}}_{\mathbf{2}}\mathbf{,}\mathrm{E}\left({\hat{\mathit{\epsilon }}}_{\mathbf{2}}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathrm{D}\left({\hat{\mathit{\epsilon }}}_{\mathbf{2}}\right)\mathbf{=}\left[\begin{array}{cc}{\mathbf{W}}_{11}& {\mathbf{W}}_{12}\\ {\mathbf{W}}_{21}& {\mathbf{W}}_{22}\end{array}\right]\triangleq \mathbf{W}\mathbf{.}

Then the simple projection prediction of linearly predictable vector Q\hat{\mathbf{y}} under model \widehat{{\mathcal{N}}_i} is

{\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{N}}_1}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)=\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\left]\right[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}{\mathbf{U}}_{\mathbf{1}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{)}\mathbf{y}\mathbf{,}

{\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{N}}_2}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)=\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\left]\right[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}{\mathbf{U}}_{\mathbf{2}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{)}\mathbf{y}\mathbf{.}

The best linear unbiased prediction of linearly predictable vector Q\hat{\mathbf{y}} under model{\mathcal{N}}_i is

{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)=\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}{\mathbf{U}}_{\mathbf{3}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{)}\mathbf{y}\mathbf{,}

{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left(\mathbf{Q}\hat{\mathbf{y}}\right)=\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\mathbf{†}}\mathbf{+}{\mathbf{U}}_{\mathbf{4}}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{)}\mathbf{y}\mathbf{.}

Assuming the true model is \widehat{{\mathcal{N}}_2}, and we mistakenly select \widehat{{\mathcal{N}}_1}, we consider the robustness of the simple projection prediction and best linear unbiased prediction of the linearly predictable vector Q\hat{\mathbf{y}} and its special cases regarding the covariance matrix under these two models.

Theorem 5.1　 Assume Q\hat{\mathbf{y}} is linearly predictable under model, then the following result is established:

(a) {\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{N}}_1}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{N}}_2}}\left(\mathbf{Q}\hat{\mathbf{y}}\right);

(b) {\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{N}}_1}}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{N}}_2}}\left(\mathbf{Q}\hat{\mathbf{y}}\right) are equal with a probability of 1;

(c) r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]=r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right];

(d) r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right];

(e) \mathcal{R}\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}{\mathbf{]}}^{\prime }\mathbf{)}\subseteq \mathcal{R}\mathbf{(}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\prime }\mathbf{)}\mathbf{.}

Proof　Through Definition 4.1(a), result (a) holds if and only if the coefficient matrices of the two projection predictors are equal:

(5.3)\begin{array}{rl}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}+{\mathbf{U}}_1{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }=\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}+{\mathbf{U}}_2{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ & \iff {\mathbf{U}}_1{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }-{\mathbf{U}}_2{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }=\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ & \iff \left[{\mathbf{U}}_1,-{\mathbf{U}}_2\right]\left[\begin{array}{c}{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right]=\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}.\end{array}

There exist U₁ and U₂ that satisfy the necessary and sufficient condition of (5.3)

(5.4)r\left[\begin{array}{c}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ [\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\\ [\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\end{array}\right]=r\left[\begin{array}{c}{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ [\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\end{array}\right].

After simplification of both sides of (5.4) through elementary transformation of (2.1) and block matrix, we can get

\begin{array}{rl}& r\left[\begin{array}{c}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ [\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\\ [\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\end{array}\right]\\ & =r\left[\begin{array}{ccc}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}& \mathbf{0}& \mathbf{0}\\ {\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ {\mathbf{I}}_n& \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & -r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]& -\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]\\ {\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ {\mathbf{I}}_n& \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\end{array}

\begin{array}{rl}& =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]& -\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]\\ {\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& -\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]\\ \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{cc}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\right]\\ \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{cccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{Q}\mathbf{Z}& \mathbf{0}\\ \mathbf{X}& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }& \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right],\end{array}

and

\begin{array}{rl}r\left[\begin{array}{c}{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ [\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\end{array}\right]& =r\left[\begin{array}{ccc}{\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ {\mathbf{I}}_n& \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & -r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{ccc}{\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left(\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right],\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\right)+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right].\end{array}

Combining the above two results, and through (2.2) and Lemma 2.2(b), we can obtain

\begin{array}{rl}& r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]=r[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp },{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }]\\ & \iff r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right],\end{array}

which indicates results (a), (c), (d), and (e) are equivalent.

From Definition 4.1(b),

{\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{M}}_1}}\left(\mathbf{Q}\hat{\mathbf{y}}\right)\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{S}\mathrm{P}\mathrm{P}}_{\widehat{{\mathcal{M}}_2}}\left(\mathbf{Q}\hat{\mathbf{y}}\right) are equal with a probability of 1

(5.5)\begin{array}{rl}& \iff \left(\right[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_1[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & -[\mathbf{Q}\mathbf{Z},\mathbf{0}][\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{†}-{\mathbf{U}}_2[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\left)\right[\mathbf{X},{\mathbf{W}}_{11}]=\mathbf{0}\\ & \iff \left(\right[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_1[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }-[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{†}\left)\right[\mathbf{X},{\mathbf{W}}_{11}]=\mathbf{0}.\end{array}

With Lemma 2.3, (5.5) is further equivalent to

(5.6)\begin{array}{rl}& \left(\right[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_1[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }-[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{†}\left)\right[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]=\mathbf{0}\\ & \iff [\mathbf{Q}\mathbf{Z},\mathbf{0}][\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]+{\mathbf{U}}_1[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]=[\mathbf{Q}\mathbf{Z},\mathbf{0}]\\ & \iff {\mathbf{U}}_1[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]=[\mathbf{Q}\mathbf{Z},\mathbf{0}]-[\mathbf{Q}\mathbf{Z},\mathbf{0}\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right].\end{array}

There exists U₁ that satisfies the necessary and sufficient condition of (5.6)

(5.7)r\left[\begin{array}{c}[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{-}\mathbf{[}\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ [\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\end{array}\right]=r\left([\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\right).

After simplification of both sides of (5.7) through elementary transformation of (2.1) and block matrix, we can get

\begin{array}{rl}& r\left[\begin{array}{c}[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{-}\mathbf{[}\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ [\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\end{array}\right]\\ & =r\left[\begin{array}{cc}[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{-}\mathbf{[}\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ [\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}& [\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{cc}[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}& [\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{0}\mathbf{]}\\ \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{cccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{Q}\mathbf{Z}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& \mathbf{X}& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right],\end{array}

and

\begin{array}{rl}r\left(\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }])& =r([\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }],[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }])-r[[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }]\\ & =r[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp },\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]-r[\mathbf{X},{\mathbf{V}}_{11}]\\ & =r[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp },{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]-r[\mathbf{X},{\mathbf{V}}_{11}].\end{array}

Combining the above results, it can be seen that (b) and (c) are equivalent. □

Theorem 5.2　 Assume Q\hat{\mathbf{y}} is linearly predictable under model {\mathcal{N}}_2, then the following results are established:

(a) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right)={\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left(\mathbf{Q}\hat{\mathbf{y}}\right);

(b) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left(\mathbf{Q}\hat{\mathbf{y}}\right) are equal with a probability of 1;

(c) r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]=r[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]};

(d) r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right];

(e) \mathcal{R}\left(\right[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }{\mathbf{]}}^{\prime }\mathbf{)}\subseteq \mathcal{R}\mathbf{(}\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\prime }\mathbf{)}\mathbf{.}

Proof　From Definition 4.1(a), result (a) holds if and only if the coefficient matrices of {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left(\mathbf{Q}\hat{\mathbf{y}}\right)) are equal, that is

(5.8)\begin{array}{rl}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}+{\mathbf{U}}_3{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ & =\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}+{\mathbf{U}}_4{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ & \iff {\mathbf{U}}_3{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }-{\mathbf{U}}_4{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ & =\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ & \iff \left[{\mathbf{U}}_3,-{\mathbf{U}}_4\right]\left[\begin{array}{c}{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right].\end{array}

There exist U₃ and U₄ that satisfy the necessary and sufficient condition of (5.8)

(5.9)r\left[\begin{array}{c}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right]=r\left[\begin{array}{c}{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right].

After simplification of both sides of (5.9) through elementary transformation of (2.1) and block matrix, we can get

\begin{array}{rl}& r\left[\begin{array}{c}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right]\\ & =r\left[\begin{array}{ccc}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}& \mathbf{0}& \mathbf{0}\\ -\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}& \mathbf{0}& \mathbf{0}\\ {\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ {\mathbf{I}}_n& \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & -r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]& -\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ {\mathbf{I}}_n& \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]& -\left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& -\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]\\ \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\end{array}

\begin{array}{rl}& =r\left[\begin{array}{cc}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]\\ \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{cccc}\mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }& \mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\\ \mathbf{X}& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }& \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& {\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }& {\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\\ \mathbf{X}& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right],\end{array}

and

\begin{array}{rl}r\left[\begin{array}{c}{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\end{array}\right]& =r\left[\begin{array}{ccc}{\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ {\mathbf{I}}_n& \mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & -r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{ccc}{\mathbf{I}}_n& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}\mathbf{0}& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\mathbf{I}}_n& \mathbf{0}& \mathbf{0}\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left(\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right],\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\right)+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right]\\ & =r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]+n-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\right].\end{array}

Combining the above two results, and with (2.2) and Lemma 2.2(b), we can obtain

\begin{array}{c}r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]=r[X,{\mathbf{V}}_{11}{\mathbf{X}}^{\perp },{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]\\ \iff r\left[\begin{array}{ccc}\mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{V}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right],\end{array}

which indicates results (a), (c), (d), and (e) are equivalent.

From Definition 4.1(b),

{\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left(\mathbf{Q}\hat{\mathbf{y}}\right) and {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left(\mathbf{Q}\hat{\mathbf{y}}\right) are equal with a probability of 1

(5.10)\begin{array}{rl}& \iff \left(\right[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_3[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & -[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{†}-{\mathbf{U}}_4[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\left)\right[\mathbf{X},{\mathbf{W}}_{11}]=\mathbf{0}\\ & \iff \left(\right[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_3[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & -[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{†})[\mathbf{X},{\mathbf{W}}_{11}]=\mathbf{0}.\end{array}

With Lemma 2.3, (5.10) is further equivalent to

(5.11)\begin{array}{rl}& \left(\right[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\left]\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}+{\mathbf{U}}_3[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }\\ & -[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }{]}^{†})[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]=\mathbf{0}\\ & \iff [\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]+{\mathbf{U}}_3[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]\\ & =[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }]\\ & \iff {\mathbf{U}}_3[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]\\ & =[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }]-[\mathbf{Q}\mathbf{Z},{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }][\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{†}[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }].\end{array}

There exists U₃ that satisfies the necessary and sufficient condition of (5.11)

(5.12)\begin{array}{rl}& r\left[\begin{array}{c}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ [\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & =r\left(\right[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }{\mathbf{]}}^{\perp }\mathbf{[}\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\mathbf{]}\mathbf{)}\mathbf{.}\end{array}

After simplification of both sides of (5.12) through elementary transformation of (2.1) and block matrix, we can obtain

\begin{array}{rl}& r\left[\begin{array}{c}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{V}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ {\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{\perp }\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & =r\left[\begin{array}{cc}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }\mathbf{X}}^{\perp }\right]-\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }\mathbf{X}}^{\perp }\right]{\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]}^{†}\left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \mathbf{0}\\ \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]\\ & -r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\\ & =r\left[\begin{array}{cc}\left[\mathbf{Q}\mathbf{Z}\mathbf{,}{\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]& \left[\mathbf{Q}\mathbf{Z}\mathbf{,}\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\right]\\ \left[\mathbf{X}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]& \left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}{\mathbf{X}}^{\perp }\right]\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{cccc}& {\mathbf{Q}\mathbf{W}\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }& \mathbf{Q}\mathbf{Z}& \mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }{\mathbf{X}}^{\perp }\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& \mathbf{X}& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right]\\ & =r\left[\begin{array}{ccc}& {\mathbf{Q}\mathbf{W}{\mathbf{T}}^{\prime }\mathbf{X}}^{\perp }& {\mathbf{Q}\mathbf{V}{\mathbf{T}}^{\prime }\mathbf{X}}^{\perp }\\ \mathbf{X}& {\mathbf{W}}_{11}{\mathbf{X}}^{\perp }& {\mathbf{V}}_{11}{\mathbf{X}}^{\perp }\end{array}\right]-r\left[\mathbf{X}\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\right],\end{array}

and

\begin{array}{rl}r\left(\right[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }{]}^{\perp }[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }])& =r([\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }],[\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }])-r[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp }]\\ & =r[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp },\mathbf{X},{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]-r[\mathbf{X},{\mathbf{V}}_{11}]\\ & =r[\mathbf{X},{\mathbf{V}}_{11}{\mathbf{X}}^{\perp },{\mathbf{W}}_{11}{\mathbf{X}}^{\perp }]-r[\mathbf{X},{\mathbf{V}}_{11}].\end{array}

Combining the above results, it can be seen that (b) and (c) are equivalent. □

Corollary 5.1　 Assume \hat{\mathbf{y}} and {\mathbf{1}}_{m+n}^{\prime }\hat{\mathbf{y}} are linearly predictable under model {\mathcal{N}}_2, then the following conclusions are established.

(a) The following results are equivalent:

(i) \mathrm{S}\mathrm{P}\mathrm{P}\widehat{{\mathcal{N}}_1}\left(\hat{\mathbf{y}}\right)=\mathrm{S}\mathrm{P}\mathrm{P}\widehat{{\mathcal{N}}_2}\left(\hat{\mathbf{y}}\right) (with a probability of 1).

(ii) r\left[\begin{array}{ccc}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right].

(b) The following results are equivalent:

(i) \mathrm{S}\mathrm{P}\mathrm{P}\widehat{{\mathcal{N}}_1}\left(1_{m+n}^{\prime }\hat{\mathbf{y}}\right)=\mathrm{S}\mathrm{P}\mathrm{P}\widehat{{\mathcal{N}}_2}\left(1_{m+n}^{\prime }\hat{\mathbf{y}}\right) (with a probability of 1).

(ii) r\left[\begin{array}{ccc}{\mathbf{1}}_{m+n}^{\prime }\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right].

Corollary 5.2　 Assume \hat{\mathbf{y}} and {\mathbf{1}}_{m+n}^{\prime }\hat{\mathbf{y}}are linearly predictable under model {\mathcal{N}}_2, and let {\mathbf{V}}_1=\left[\begin{array}{c}{\mathbf{V}}_{11}\\ {\mathbf{V}}_{21}\end{array}\right], {\mathbf{W}}_1=\left[\begin{array}{c}{\mathbf{W}}_{11}\\ {\mathbf{W}}_{21}\end{array}\right]. Then the following conclusions hold.

(a) The following two conditions are equivalent:

(i) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left(\hat{\mathbf{y}}\right)={\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left(\hat{\mathbf{y}}\right)(with a probability of 1).

(ii) r\left[\begin{array}{ccc}\mathbf{Z}& {\mathbf{W}}_1& {\mathbf{V}}_1\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right].

(b) The following two conditions are equivalent:

(i) {\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_1}\left({\mathbf{1}}_{m+n}^{\prime }\hat{\mathbf{y}}\right)={\mathrm{B}\mathrm{L}\mathrm{U}\mathrm{P}}_{{\mathcal{N}}_2}\left({\mathbf{1}}_{m+n}^{\prime }\hat{\mathbf{y}}\right)(with a probability of 1).

(ii) r\left[\begin{array}{ccc}{1}_{m+n}^{\prime }\mathbf{Z}& {\mathbf{1}}_n^{\prime }{\mathbf{W}}_{11}+{\mathbf{1}}_{\mathit{m}}^{\prime }{\mathbf{W}}_{21}& {\mathbf{1}}_n^{\prime }{\mathbf{V}}_{11}+{\mathbf{1}}_m^{\prime }{\mathbf{V}}_{21}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right].

Next, we use the error uniform correlation models in statistics to illustrate the specific application of equivalent conditions.

Assume

\widehat{{\mathcal{N}}_1}:\hat{\mathbf{y}}={\mathbf{1}}_{m+n}\mathit{\beta }+{\hat{\mathit{\epsilon }}}_1,\mathrm{E}\left({\hat{\mathit{\epsilon }}}_1\right)=\mathbf{0},\mathrm{D}\left({\hat{\mathit{\epsilon }}}_1\right)={\sigma }^2{\mathrm{I}}_{m+n},

\widehat{{\mathcal{M}}_2}:\hat{\mathbf{y}}={\mathbf{1}}_{m+n}\mathit{\beta }+{\hat{\mathit{\epsilon }}}_2,\mathrm{E}\left({\hat{\mathit{\epsilon }}}_2\right)=\mathbf{0},\mathrm{D}\left({\hat{\mathit{\epsilon }}}_2\right)={\sigma }^2\left[\begin{array}{cccc}1& \rho & \cdots & \rho \\ \rho & 1& \cdots & \rho \\ ⋮& ⋮& \ddots & ⋮\\ \rho & \rho & \cdots & 1\end{array}\right],

where {\mathbf{1}}_{m+n}=[1,1,\dots ,1{]}^{\prime } are vectors of (m+n)×1 order with all components being 1, and I_m+nare unit matrices of m+n order. The covariance matrix of model \widehat{{\mathcal{N}}_2} can be rewritten as

\widehat{{\mathcal{N}}_2}:\hat{\mathbf{y}}={\mathbf{1}}_{m+n}\mathit{\beta }+{\hat{\mathit{\epsilon }}}_2,\mathrm{E}\left({\hat{\mathit{\epsilon }}}_2\right)=\mathbf{0},\mathrm{D}\left({\hat{\mathit{\epsilon }}}_2\right)={\sigma }^2[\rho {\mathbf{1}}_{m+n}{\mathbf{1}}_{m+n}^{\prime }+(\mathbf{1}-\rho \left){\mathbf{I}}_{m+n}\right].

That is, there is equal variance {\sigma }^2 in all observations, and the same correlation coefficient as well. Assume \mathbf{Q}\mathbf{=}{\mathbf{I}}_{\mathbf{m}\mathbf{+}\mathbf{n}}, and \mathbf{Q}\hat{\mathbf{y}}\mathbf{=}\hat{\mathbf{y}} is linearly predictable under model {\mathcal{N}}_2,

\mathbf{Z}\mathbf{=}{\mathbf{1}}_{\mathbf{m}\mathbf{+}\mathbf{n}}\mathbf{,}\mathbf{X}\mathbf{=}{\mathbf{1}}_{\mathbf{n}}\mathbf{,}{\mathbf{W}}_{\mathbf{11}}\mathbf{=}{\sigma }^{\mathbf{2}}\mathbf{[}\rho {\mathbf{1}}_{\mathbf{n}}{\mathbf{1}}_{\mathbf{n}}^{\prime }\mathbf{+}\mathbf{(}\mathbf{1}\mathbf{-}\rho \left){\mathbf{I}}_{\mathbf{n}}\right]\mathbf{,}{\mathbf{V}}_{\mathbf{11}}\mathbf{=}{\sigma }^{\mathbf{2}}{\mathbf{I}}_{\mathbf{n}}\mathbf{.}

From the equivalent condition (ii) in Corollary 5.1(a), the left side of the equation is

\begin{array}{rl}r\left[\begin{array}{ccc}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]& =r\left[\begin{array}{ccc}{\mathbf{1}}_{m+n}& \mathbf{0}& \mathbf{0}\\ {\mathbf{1}}_n& {\sigma }^2[\rho {\mathbf{1}}_n{\mathbf{1}}_n^{\prime }+(1-\rho \left){\mathbf{I}}_n\right]& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{1}}_{\mathit{n}}^{\prime }\end{array}\right]\\ & =r\left[\begin{array}{ccc}{\mathbf{1}}_{m+n}& \mathbf{0}& \mathbf{0}\\ {\mathbf{1}}_n& {\sigma }^2\rho {\mathbf{1}}_n{\mathbf{1}}_n^{\prime }& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ \mathbf{0}& (\rho -1){\mathbf{1}}_n^{\prime }& {\mathbf{1}}_n^{\prime }\end{array}\right]\\ & =r\left[\begin{array}{ccc}{\mathbf{1}}_{m+n}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{1}}_n^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}{\mathbf{1}}_{m+n}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\end{array}\right]=n+2,\end{array}

and the right side of the equation is

\begin{array}{rl}r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]& =r\left[\begin{array}{ccc}{\mathbf{1}}_n& {\sigma }^2[\rho {\mathbf{1}}_n{\mathbf{1}}_n^{\prime }+(1-\rho \left){\mathbf{I}}_n\right]& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{1}}_n^{\prime }\end{array}\right]\\ & =r\left[\begin{array}{ccc}{\mathbf{1}}_n& {\sigma }^2\rho {\mathbf{1}}_n{\mathbf{1}}_n^{\prime }& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ \mathbf{0}& (\rho -1){\mathbf{1}}_n^{\prime }& {\mathbf{1}}_n^{\prime }\end{array}\right]\\ & =r\left[\begin{array}{ccc}{\mathbf{1}}_n& \mathbf{0}& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ {\sigma }^{-2}n& \mathbf{0}& \mathbf{0}\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{0}& \mathbf{0}& {\sigma }^2{\mathbf{I}}_n\\ \mathbf{0}& {\mathbf{1}}_n^{\prime }& \mathbf{0}\\ {\sigma }^{-2}n& \mathbf{0}& \mathbf{0}\end{array}\right]=n+2.\end{array}

Hence

r\left[\begin{array}{ccc}\mathbf{Z}& \mathbf{0}& \mathbf{0}\\ \mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right]=r\left[\begin{array}{ccc}\mathbf{X}& {\mathbf{W}}_{11}& {\mathbf{V}}_{11}\\ \mathbf{0}& {\mathbf{X}}^{\prime }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{X}}^{\prime }\end{array}\right].

Then, the equivalent condition (ii) in Corollary 5.1(a) is satisfied, namely the SPP is robust under this model.

6 Conclusions

SPP and BLUP are two very basic predictors of unknown parameters in linear models. SPP boasts a simple form and is very convenient to use in statistical inference; BLUP, on the other hand, is the predictor with the smallest covariance among all linear unbiased predictors of unknown parameters under Löwner partial order. It is often used as a comparison standard for the soundness of other predictors for excellent algebraic and statistical properties. The study of the relationship between these two predictors is an important work. This paper mainly uses a new method - the basic block matrix rank formula in matrix algebra theory, i.e. the matrix rank and block matrix method, to provide various necessary and sufficient conditions for the rank and spatial inclusion relationship between them. Finally, the robustness of SPP and BLUP with respect to the covariance matrix is studied separately, and specific applications of equivalent conditions are illustrated with error uniform correlation models in statistics, lending important theoretical references for subsequent statistical inference and research on predictor relationships under other complex statistical models.

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