A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications

Xingwei Ren

doi:10.1515/dema-2025-0152

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A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications

Published/Copyright: July 29, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

Suppose that a weakly singular linear regression model M and its two competing restricted models M 1 and M 2 are given. We first compare the superiorities of the best linear unbiased estimators (BLUEs) under M and M 1 related to the covariance matrix criterion and derive some necessary and sufficient conditions. Based on these conditions and the covariance matrix criterion, we then compare the superiorities of the BLUEs under M 1 and M 2 and obtain some results. We finally apply these results to measure the influence of an observation on the BLUE.

Keywords: weakly singular linear regression model; restricted model; superiority; BLUE

MSC 2010: 15A09; 62H12; 62J05

1 Introduction

Throughout this article, M ′ , M † , r ( M ) , and C ( M ) , respectively, denote its transpose, Moore-Penrose inverse, rank, and column space for a matrix M and R m × n stands for the collection of all m × n real matrices. Define i + ( M ) and i − ( M ) as the number of the positive and negative eigenvalues of a symmetric matrix M computed with multiplicities respectively, and postulate P M = MM † , E M = I − MM † , and F M = I − M † M . Moreover, for symmetric matrices M ∈ R n × n and N ∈ R n × n , M ≻ N ( ≽ 0 , ≺ 0 , ≼ 0 ) in the Löwner partial ordering means that i + ( M − N ) = n ( i − ( M − N ) = 0 , i − ( M − N ) = n , i + ( M − N ) = 0 ) . For briefness, we use M = { y , X β , Σ } to denote the regression model

(1.1) y = X β + ε , E ( ε ) = 0 , Cov ( ε ) = Σ ≽ 0 ,

where y ∈ R n × 1 is an observable random vector, β ∈ R p × 1 is a parameter vector, ε ∈ R n × 1 is a vector of unobservable disturbances, and X ∈ R n × p and Σ are two known matrices without any rank restrictions.

Let two competing stochastic restrictions be added on regression parameter β that may be shown by

(1.2) r j = A j β + e j , E ( e j ) = 0 , Cov ( e j ) = V j , j = 1 , 2 ,

where e j ∈ R m j × 1 is a random error vector such that Cov ( ε , e j ) = 0 , A j ∈ R m j × p , V j ∈ R m j × m j , and r j ∈ R m j × 1 are three known matrices with any rank. The constant vector r j may be approached as a random vector with E ( r j ) = A j β , j = 1, 2 [1, p. 252].

Linear stochastic restrictions may come from former studies or some external source and have numerous applications to deletion diagnostics in regression analysis [2,3], bioassay in biometric research [4], economic relations [5,6], among others. Alternatively, we can apply linear stochastic restrictions to the estimation of linear regression models with missing data [7]. Furthermore, as stated by Arashi and Tabatabaey [8], by linear stochastic restrictions, we can implement an analysis and examination of one’s own feelings and thoughts. Additionally, integrating linear stochastic restrictions and Zellner’s [9] “Seemingly Unrelated” estimators, we can comparatively reveal good properties of estimators using a mean squared error criterion. For the impact of linear stochastic restrictions on estimators and predictors, see [10–14], for the whole review on linear stochastic restrictions and applications, see [5].

As proposed first by Theil and Goldberger [15], using the regression model (1.1) and stochastic restrictions (1.2) simultaneously yields the following common model:

(1.3) M j = y r j , X β A j β , Σ 0 0 V j , j = 1 , 2 .

When r ( X ) = p , Σ ≻ 0 , and V j ≻ 0 , j = 1 , 2 , it is well known that the best linear unbiased estimator (BLUE) of β in (1.3) can be written as

BLUE ( β ∣ M j ) = ( S + A j ′ V j − 1 A j ) − 1 ( X ′ Σ − 1 y + A j ′ V j − 1 r j ) = BLUE ( β ∣ M ) + S − 1 A j ′ ( V j + A j S − 1 A j ′ ) − 1 ( r j − A j BLUE ( β ∣ M ) ) , j = 1 , 2 ,

where S = X ′ Σ − 1 X and BLUE ( β ∣ M ) = ( X ′ Σ − 1 X ) − 1 X ′ Σ − 1 y is the BLUE of β in (1.1). By straightforward operations, we obtain

Cov ( BLUE ( β ∣ M ) ) ≽ Cov ( BLUE ( β ∣ M j ) ) , j = 1 , 2 .

In 2007, Xu and Yang generalized the above result to a certain extent, obtaining a more general situation. However, by further research, we find that the conclusion may be further enhanced and is presented in the next section. Suppose that X has a full column rank, i.e., r ( X ) = p , Σ = I n , and r ( A j ) = m j , j = 1 , 2 , in (1.3), Baksalary [16] considered the problem of making comparison of BLUEs of parameter β under M 1 and M 2 using the covariance matrix criterion, as well as [17,18]. The resulting necessary and sufficient conditions were deduced, which suggested that one of the estimators outperformed the other. While in many cases, the assumption mentioned above is not reasonable, as in (3.6). To obtain a more general result, we extend the linear regression models in [16–23] to a weakly singular linear regression model, i.e., the model in (1.1) satisfies C ( X ) ⊆ C ( Σ ) . The main contribution of this work is to construct exact formulas of the comparisons between BLUEs of X β under the models M 1 and M 2 with C ( X ) ⊆ C ( Σ ) relative to their covariance matrices utilizing matrix rank and index of inertia methods, as well as between BLUEs of β . Through these exact formulas, it is quite convenient to make comparison between two BLUEs since the rank and index of inertia of a matrix can be readily checked in practice. In this article, our approach to treating these two competing restriction equations (1.2) is available for the research area in a biological assay. For instance, in slope ratio assays, it is necessary to consider the cases of imposing the stochastic restrictions (1.2) on regression parameters [4]. Moreover, another reason for considerations on restrictions (1.2) is for specialized persons to make adjustment tests based on the comparison of estimators under M 1 and M 2 , for example, to discard one of two restrictions (1.2).

Now, let us recollect the definition that if Gy is an unbiased estimator of X β , then

(1.4) Gy = BLUE ( X β ∣ M ) ⇔ Cov ( Gy ) ≼ Cov ( Hy ) ∀ H : HX = X ⇔ G ( X , Σ E X ) = ( X , 0 ) ,

see [19]. The solution to (1.4) is

(1.5) G = ( X , 0 ) ( X , Σ E X ) † + U E ( X , Σ E X ) ,

where U ∈ R n × n is an arbitrary matrix.

Hence, in terms of (1.4) and (1.5), it can be obtained that

(1.6) BLUE ( X β ∣ M j ) = G j y r j , j = 1 , 2 ,

where

(1.7) G j = ( X,0 ) ( X ^ j , Σ ^ j E X ^ j ) † + U j E ( X ^ j , Σ ^ j E X ^ j ) ,

in which X ^ j = X A j , Σ ^ j = Σ 0 0 V j , , and U j ∈ R n × ( n + m j ) is an arbitrary matrix.

For the sake of simplification of our conclusions including the Moore-Penrose inverses of matrices, we assemble the following three results furnished by [20].

Lemma 1.1

Let Q 1 ∈ R n × n and Q 2 ∈ R m × m be both symmetric matrices and Q 3 ∈ R n × m . Then

(1.8) i ± ( P Q 1 P ′ ) ≼ i ± ( Q 1 ) i f P ∈ R n × n i s a n a r b i t r a r y m a t r i x ,

(1.9) i ± ( P Q 1 P ′ ) = i ± ( Q 1 ) i f P ∈ R n × n i s a n o n s i n g u l a r m a t r i x ,

(1.10) i ± Q 1 0 0 Q 2 = i ± ( Q 1 ) + i ± ( Q 2 ) ,

(1.11) i ± 0 Q 3 Q 3 ′ 0 = r ( Q 3 ) , i ± ( − Q 1 ) = i ∓ ( Q 1 ) .

Lemma 1.2

Let Q 1 ∈ R n × n be a symmetric matrix and let Q 2 ∈ R n × m and Q 3 ∈ R m × m . Then

(1.12) i ± Q 1 Q 2 Q 2 ′ 0 = r ( Q 2 ) + i ± ( E Q 2 Q 1 E Q 2 ) ,

(1.13) i + Q 1 Q 2 Q 2 ′ 0 = r ( Q 1 , Q 2 ) a n d i − Q 1 Q 2 Q 2 ′ 0 = r ( Q 2 ) , i f Q 1 ≽ 0 ,

(1.14) i + Q 1 Q 2 Q 2 ′ 0 = r ( Q 2 ) a n d i − Q 1 Q 2 Q 2 ′ 0 = r ( Q 1 , Q 2 ) , i f Q 1 ≼ 0 ,

(1.15) i ± Q 1 Q 2 Q 2 ′ Q 3 ≽ i ± ( Q 1 ) .

Lemma 1.3

Assume that Q j ∈ R n j × n j is a symmetric matrix and let O j ∈ R l j × m and P j ∈ R l j × n j , j = 1 , 2 . If C ( Q j ) ⊆ C ( P j ′ ) and C ( O j ) ⊆ C ( P j ) , j = 1 , 2 , then

(1.16) i ± [ O 1 ′ ( P 1 ′ ) † Q 1 P 1 † O 1 − O 2 ′ ( P 2 ′ ) † Q 2 P 2 † O 2 ] = i ± Q 1 0 P 1 ′ 0 0 0 − Q 2 0 P 2 ′ 0 P 1 0 0 0 O 1 0 P 2 0 0 O 2 0 0 O 1 ′ O 2 ′ 0 − r ( P 1 ) − r ( P 2 ) .

We also need the following lemma supplied by Marsaglia and Styan [21].

Lemma 1.4

Let Q 1 ∈ R m × n , Q 2 ∈ R m × k , and Q 3 ∈ R l × n . Then

(1.17) r ( Q 1 , Q 2 ) = r ( Q 1 ) + r ( E Q 1 Q 2 ) = r ( Q 2 ) + r ( E Q 2 Q 1 ) ,

(1.18) r Q 1 Q 3 = r ( Q 1 ) + r ( Q 3 F Q 1 ) = r ( Q 3 ) + r ( Q 1 F Q 3 ) ,

(1.19) r Q 1 Q 2 Q 3 0 = r ( Q 2 ) + r ( Q 3 ) + r ( E Q 2 Q 1 F Q 3 ) .

2 Comparing the BLUEs

In this section, we first compare covariance matrices of BLUEs under M and M 1 . On the basis of the resulting consequence, we then make comparison of covariance matrices of BLUEs under M 1 and M 2 . We observe by (1.4)–(1.7) that

(2.1) Cov ( BLUE ( X β ∣ M ) ) = G Σ G ′ = ( X,0 ) ( X , Σ E X ) † Σ [ ( X , Σ E X ) ′ ] † ( X , 0 ) ′ ,

(2.2) Cov ( BLUE ( X β ∣ M j ) ) = G j Σ ^ j G j ′ = ( X,0 ) ( X ^ j , Σ ^ j E X ^ j ) † Σ ^ j [ ( X ^ j , Σ ^ j E X ^ j ) ′ ] † ( X,0 ) ′ , j = 1 , 2 .

Assumption 2.1

Suppose that M is a weakly singular linear regression model, namely, C ( X ) ⊆ C ( Σ ) , as depicted in the literature.

Theorem 2.1

Consider the models M and M 1 under Assumption 2.1 and define BLUE ( β ∣ M ) and BLUE ( β ∣ M 1 ) as the BLUEs of β under M and M 1 , respectively. Then,

Cov ( BLUE ( X β ∣ M 1 ) ) ≼ Cov ( BLUE ( X β ∣ M ) ) .
Cov ( BLUE ( β ∣ M 1 ) ) ≼ Cov ( BLUE ( β ∣ M ) ) .
Cov ( BLUE ( X β ∣ M 1 ) ) ≺ Cov ( BLUE ( X β ∣ M ) ) if and only if r ( A ) + r ( X ) − r X A 1 = n .
Cov ( BLUE ( X β ∣ M 1 ) ) = Cov ( BLUE ( X β ∣ M ) ) if and only if r ( A ) + r ( X ) = r X A 1 , or equivalently C ( A 1 ′ ) ∩ C ( X ′ ) = { 0 } .
Cov ( BLUE ( β ∣ M 1 ) ) ≺ Cov ( BLUE ( β ∣ M ) ) if and only if r ( A 1 ) = p .
Cov ( BLUE ( β ∣ M 1 ) ) = Cov ( BLUE ( β ∣ M ) ) if and only if A 1 = 0 .

Proof

Applying Lemma 1.3 to the difference between (2.1) and (2.2) for j = 1 leads to

(2.3) i ± ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M ) ) ) = i ± ( ( X , 0 ) ( X ^ 1 , Σ ^ 1 E X ^ 1 ) † Σ ^ 1 [ ( X ^ 1 , Σ ^ 1 E X ^ 1 ) ′ ] † ( X,0 ) ′ − ( X,0 ) ( X , Σ E X ) † Σ [ ( X , Σ E X ) ′ ] † ( X , 0 ) ′ ) = i ± Σ ^ 1 0 X ^ 1 Σ ^ 1 E X ^ 1 0 0 0 0 − Σ 0 0 X Σ E X 0 X ^ 1 ′ 0 0 0 0 0 X ′ E X ^ 1 Σ ^ 1 0 0 0 0 0 0 0 X ′ 0 0 0 0 X ′ 0 E X Σ 0 0 0 0 0 0 0 X 0 X 0 0 − r ( X ^ 1 , Σ ^ 1 E X ^ 1 ) − r ( X , Σ E X ) .

Noticing r ( X , Σ E X ) = r ( X , Σ ) and r ( X ^ 1 , Σ ^ 1 E X ^ 1 ) = r ( X ^ 1 , Σ ^ 1 ) and utilizing (1.9) and (1.10), (2.3) can be equivalently written as

(2.4) i ± Σ ^ 1 0 X ^ 1 0 0 0 0 0 − Σ 0 0 X 0 0 X ^ 1 ′ 0 0 0 0 0 X ′ 0 0 0 − E X ^ 1 Σ ^ 1 E X ^ 1 0 0 0 0 X ′ 0 0 0 0 X ′ 0 0 0 0 0 E X Σ E X 0 0 0 X 0 X 0 0 − r ( X ^ 1 , Σ ^ 1 ) − r ( X , Σ ) = i ± Σ ^ 1 0 X ^ 1 0 0 0 − Σ 0 X 0 X ^ 1 ′ 0 0 0 X ′ 0 X ′ 0 0 X ′ 0 0 X X 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X Σ E X ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X , Σ ) = i ± Σ ^ 1 0 X ^ 1 0 0 0 − Σ − X 0 0 X ^ 1 ′ − X ′ 0 0 0 0 0 0 0 X ′ 0 0 0 X 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X Σ E X ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X , Σ ) .

First, using (1.10) and (1.11) and then using Assumption 2.1, we deduce that (2.4) is

i ± Σ 0 0 X 0 V 1 0 A 1 0 0 − Σ − X X ′ A 1 ′ − X ′ 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X Σ E X ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X , Σ ) + r ( X ) = i ± Σ 0 0 0 0 V 1 0 A 1 0 0 − Σ 0 0 A 1 ′ 0 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X Σ E X ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X , Σ ) + r ( X ) = i ± V 1 A 1 A 1 ′ 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X Σ E X ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X , Σ ) + r ( X ) + r ( Σ ) .

In consideration of

C ( X ) ⊆ C ( Σ ) , r ( X ^ 1 , Σ ^ 1 E X ^ 1 ) = r ( X ^ 1 ) + r ( Σ ^ 1 E X ^ 1 ) , r ( X , Σ E X ) = r ( X ) + r ( Σ E X )

and (1.13), we have

(2.5) i + ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M ) ) ) = 0 ,

(2.6) i − ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M ) ) ) = r ( X ) + r ( A 1 ) − r X A 1 ,

which gives rise to (a). If we let β be estimable under M , then according to the definition of estimability, there is an invertible matrix Q ∈ R n × n satisfying

(2.7) QX = ( Q 1 ′ , Q 2 ′ ) ′ ,

where Q 1 ∈ R p × p is a nonsingular matrix. Put

(2.8) Ω = Cov ( BLUE ( β ∣ M 1 ) ) − Cov ( BLUE ( β ∣ M ) ) .

We have

(2.9) Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M ) ) = X Ω X ′ .

Utilizing (2.7) and applying (1.8), (1.9), and (1.15) yield the subsequent relationship

(2.10) i ± ( Ω ) ≽ i ± ( X Ω X ′ ) = i ± ( Q X Ω X ′ Q ′ ) = i ± Q 1 Ω Q 1 ′ Q 1 Ω Q 2 ′ Q 2 Ω Q 1 ′ Q 2 Ω Q 2 ′ ≽ i ± ( Q 1 Ω Q 1 ′ ) = i ± ( Ω ) ,

i.e.,

(2.11) i ± ( Ω ) = i ± ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M ) ) ) ,

which, by (2.5), implies (b). Apparently, (c)–(f) hold by (2.5) and (2.6)□

Theorem 2.1 (b) is the generalized form of Theorem 3.1 in [11]. It is noteworthy that (1.6) includes covariance matrix V 1 , while the results in Theorem 2.1 have nothing to do with it. And the conclusions (a) and (b) in Theorem 2.1 are quite intuitive, because more information is added to the unknown regression parameters, which brings about a higher accuracy for the corresponding estimators. In addition, Ren [13] showed that

BLUE ( X β ∣ M 1 ) = BLUE ( X β ∣ M )

with probability 1 if and only if

C ( A 1 ′ ) ∩ C ( X ′ ) = { 0 }

with Assumption 2.1, which is compared with (d) in Theorem 2.1.

For the restriction models M 1 and M 2 meeting A 1 = A 2 , some authors explored the superiority of BLUE over the other with some assumptions to these two restriction models, which is related to covariance matrix criterion, and showed that there exists a one-to-one match between the covariance matrix of random error vector in (1.2) and BLUE under the restriction model in (1.3) [16]. In addition to the case adding A 1 = A 2 on M 1 and M 2 , there was also some literature handling how to compare the dominance of BLUEs under M 1 and M 2 related to the covariance matrix criterion when the assertion A 1 ≠ A 2 occurs, see [17,18]. However, when faced with considerable generalized inverses of matrices included in BLUEs under M 1 and M 2 , statisticians had no proper methods to take up them and as a result did not give a more general conclusion on the previous problem in the existing literature. Fortunately, with the development of matrix rank and index of inertia method we can pay attention to addressing this problem. For more details on the matrix index of inertia method, see [20]; for more details on the matrix rank method, see [21]; for the applications of matrix rank and index of inertia methods in statistical inference, see [22–31]. When we wish to compare the BLUEs under M 1 and M 2 in the sense of the covariance matrix criterion, the following assumption, in terms of conclusion (d) in Theorem 2.1, may be introduced.

Assumption 2.2

C ( A j ′ ) ⊂ C ( X ′ ) , or equivalently r ( X ′ , A j ′ ) = r ( X ) , j = 1 , 2 .

Theorem 2.2

Consider these two models M 1 and M 2 appearing in (1.3) under Assumptions 2.1 and 2.2. Then,

(a) The inequality Cov ( BLUE ( X β ∣ M 1 ) ) ≼ Cov ( BLUE ( X β ∣ M 2 ) ) holds if and only if

(2.12) i + V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 = r ( V 1 , A 1 ) .

(b) The inequality Cov ( BLUE ( X β ∣ M 1 ) ) ≺ Cov ( BLUE ( X β ∣ M 2 ) ) holds if and only if

(2.13) i − V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 = r ( V 2 , A 2 ) + n .

(2.14) r V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 = r ( V 1 , A 1 ) + r ( V 2 , A 2 ) .

Proof

The application of Lemma 1.3 to the difference between covariances of BLUEs for X β under M 1 and M 2 yields

i ± ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M 2 ) ) ) = i ± ( ( X , 0 ) ( X ^ 1 , Σ ^ 1 E X ^ 1 ) † Σ ^ 1 [ ( X ^ 1 , Σ ^ 1 E X ^ 1 ) ′ ] † ( X,0 ) ′ − ( X , 0 ) ( X ^ 2 , Σ ^ 2 E X ^ 2 ) † Σ ^ 2 [ ( X ^ 2 , Σ ^ 2 E X ^ 2 ) ′ ] † ( X,0 ) ′ ) = i ± Σ ^ 1 0 X ^ 1 Σ ^ 1 E X ^ 1 0 0 0 0 − Σ ^ 2 0 0 X ^ 2 Σ ^ 2 E X ^ 2 0 X ^ 1 ′ 0 0 0 0 0 X ′ E X ^ 1 Σ ^ 1 0 0 0 0 0 0 0 X ^ 2 ′ 0 0 0 0 X ′ 0 E X ^ 2 Σ ^ 2 0 0 0 0 0 0 0 X 0 X 0 0 − r ( X ^ 1 , Σ ^ 1 E X ^ 1 ) − r ( X ^ 2 , Σ ^ 2 E X ^ 2 ) ,

which dealed with by (1.9) and r ( X ^ j , Σ ^ j E X ^ j ) = r ( X ^ j , Σ ^ j ) , j = 1 , 2 , results in

(2.15) i ± Σ ^ 1 0 X ^ 1 0 0 0 0 0 − Σ ^ 2 0 0 X ^ 2 0 0 X ^ 1 ′ 0 0 0 0 0 X ′ 0 0 0 − E X ^ 1 Σ ^ 1 E X ^ 1 0 0 0 0 X ^ 2 ′ 0 0 0 0 X ′ 0 0 0 0 0 E X ^ 2 Σ ^ 2 E X ^ 2 0 0 0 X 0 X 0 0 − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) .

Applying (1.10) to (2.15) and simplifying by (1.11) lead to

(2.16) i ± Σ ^ 1 0 X ^ 1 0 0 0 − Σ ^ 2 0 X ^ 2 0 X ^ 1 ′ 0 0 0 X ′ 0 X ^ 2 ′ 0 0 X ′ 0 0 X X 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X ^ 2 Σ ^ 2 E X ^ 2 ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) = i ± Σ ^ 1 0 X ^ 1 0 0 0 − Σ ^ 2 − X ^ 2 0 0 X ^ 1 ′ − X ^ 2 ′ 0 0 0 0 0 0 0 X ′ 0 0 0 X 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X ^ 2 Σ ^ 2 E X ^ 2 ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) = i ± Σ ^ 1 0 X ^ 1 0 − Σ ^ 2 − X ^ 2 X ^ 1 ′ − X ^ 2 ′ 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X ^ 2 Σ ^ 2 E X ^ 2 ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) + r ( X ) = i ± Σ 0 0 0 X 0 V 1 0 0 A 1 0 0 − Σ 0 − X 0 0 0 − V 2 − A 2 X ′ A 1 ′ − X ′ − A 2 ′ 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X ^ 2 Σ ^ 2 E X ^ 2 ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) + r ( X ) .

In light of Assumption 2.1, we obtain

(2.17) i ± Σ 0 0 0 0 0 V 1 0 0 A 1 0 0 − Σ 0 0 0 0 0 − V 2 − A 2 0 A 1 ′ 0 − A 2 ′ 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X ^ 2 Σ ^ 2 E X ^ 2 ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) + r ( X ) = i ± V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 + i ∓ ( E X ^ 1 Σ ^ 1 E X ^ 1 ) + i ± ( E X ^ 2 Σ ^ 2 E X ^ 2 ) − r ( X ^ 1 , Σ ^ 1 ) − r ( X ^ 2 , Σ ^ 2 ) + r ( X ) + r ( Σ ) .

Therefore, by Assumption 2.2, we give

i + ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M 2 ) ) ) = i + V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 − r ( V 1 , A 1 ) , i − ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M 2 ) ) ) = i − V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 − r ( V 2 , A 2 ) ,

establishing the desired results (a)–(c).□

Due to the non-negative definiteness of V j , i.e., V j ≽ 0 , there exists an invertible matrix P j such that

P j V j P j ′ = I l j 0 0 0

with r ( V j ) = l j , j = 1 , 2 . Apparently,

r j = A j β + e j ⇔ r ˜ j = A ˜ j β + e ˜ j ,

where r ˜ j = P j r j , A ˜ j = P j A j and e ˜ j = P j e j , j = 1 , 2 . In regard of

Cov ( e ˜ j ) = I l j 0 0 0 , j = 1 , 2 ,

the following statement may be postulated.

Assumption 2.3

V j = I l j 0 0 0 and A j = A j 1 A j 2 l j m j − l j , 0 ≤ l j ≤ m j , j = 1 , 2 , where l j and m j − l j stand for the number of the rows in A j 1 and A j 2 , respectively.

Under Assumption 2.3, Theorem 2.2 can be reduced to the following statement.

Corollary 2.1

Consider both models M 1 and M 2 appearing in (1.3) under Assumptions 2.1 and 2.3.

(a) The inequality Cov ( BLUE ( X β ∣ M 1 ) ) ≼ Cov ( BLUE ( X β ∣ M 2 ) ) holds if and only if

(2.18) i + A 21 ′ A 21 − A 11 ′ A 11 A 12 ′ A 22 ′ A 12 0 0 A 22 0 0 = r ( A 12 ) ,

or equivalently,

(2.19) C ( A 22 ′ ) ⊂ C ( A 12 ′ ) and F A 12 ( A 21 ′ A 21 − A 11 ′ A 11 ) F A 12 ≤ 0 .

(b) The inequality Cov ( BLUE ( X β ∣ M 1 ) ) ≺ Cov ( BLUE ( X β ∣ M 2 ) ) holds if and only if

(2.20) i − A 21 ′ A 21 − A 11 ′ A 11 A 12 ′ A 22 ′ A 12 0 0 A 22 0 0 = r ( A 22 ) + n .

(2.21) r A 21 ′ A 21 − A 11 ′ A 11 A 12 ′ A 22 ′ A 12 0 0 A 22 0 0 = r ( A 12 ) + r ( A 22 ) ,

or equivalently,

(2.22) C ( A 22 ′ ) = C ( A 12 ′ ) and F A 12 ( A 21 ′ A 21 − A 11 ′ A 11 ) F A 12 = 0 .

Proof

Note from Assumption 2.3 that

(2.23) r ( V 1 , A 1 ) = r ( A 12 ) + l 1

and

(2.24) i + V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 = i + A 21 ′ A 21 − A 11 ′ A 11 A 12 ′ A 22 ′ A 12 0 0 A 22 0 0 + l 1 .

Therefore, (2.12) is precisely (2.18). Also, observe from Assumption 2.3 that

(2.25) r ( V 2 , A 2 ) = r ( A 22 ) + l 2

and

(2.26) i − V 1 0 A 1 0 − V 2 A 2 A 1 ′ A 2 ′ 0 = i − A 21 ′ A 21 − A 11 ′ A 11 A 12 ′ A 22 ′ A 12 0 0 A 22 0 0 + l 2 ,

which implies (2.20). Substituting (2.23)–(2.26) into (2.14) leads to (2.21). Note that E A 12 ′ = F A 12 and from C ( A 22 ′ ) ⊂ C ( A 12 ′ ) consequently E ( A 12 ′ , A 22 ′ ) = F A 12 . Hence, by (1.12), the equivalence (2.18) ⇔ (2.19) is established. Likewise, we can obtain (2.21) ⇔ (2.22).□

Let β be estimable under M 1 and M 2 , that is,

(2.27) r ( X ′ , A 1 ′ ) = r ( X ′ , A 2 ′ ) = p .

The application of Assumption 2.2 to (2.27) gives r ( X ) = p . Analogous to (2.7)–(2.11) of Theorem 2.1, it is demonstrated that

i ± ( Ω ) = i ± ( Cov ( BLUE ( X β ∣ M 1 ) ) − Cov ( BLUE ( X β ∣ M 2 ) ) ) ,

where Ω = Cov ( BLUE ( β ∣ M 1 ) ) − Cov ( BLUE ( β ∣ M 2 ) ) , and hence, it can be correspondingly acquired that necessary and sufficient conditions for the superiority of the BLUE for β over the other under M 1 and M 2 relative to the covariance matrix criterion. Assume that (1.2) are both exact linear restrictions, i.e., V j = 0 , j = 1 , 2 . In this case, the comparison of covariance matrices of BLUEs for β under M 1 and M 2 was first investigated by Trenkler [32], then extended by Ren and Zhou [33], which is the special situation of the above conclusion. It should be emphasized that the easier conditions in corollary 2.1 can be found once l 1 and l 2 are specified, for instance, if l 1 = m 1 and l 2 = m 2 in Assumption 2.3, then corollary 2.1 is simplified as follows.

Corollary 2.2

Under Assumptions 2.1 and 2.3, and let l 1 = m 1 and l 2 = m 2 in Assumption 2.3. Then,

(a) Cov ( BLUE ( X β ∣ M 1 ) ) ≼ Cov ( BLUE ( X β ∣ M 2 ) ) ⇔ A 2 ′ A 2 − A 1 ′ A 1 ≼ 0 .

(b) Cov ( BLUE ( X β ∣ M 1 ) ) ≺ Cov ( BLUE ( X β ∣ M 2 ) ) ⇔ i − ( A 2 ′ A 2 − A 1 ′ A 1 ) = n .

3 Applications

There are a lot of ways to measure the influence of an observation on estimators in the literature, see, e.g. [34] for Cook’s distance, [35] for Welsch-Kuh’s distance, and [36] for Andrews-Pregibon statistic, etc. Set

(3.1) X ^ j ′ X ^ j ≻ 0 , j = 1 , 2 ,

m 1 = m 2 = 1 in (1.2), and then regard (1.2) as two additional observations. In the model with n + 2 observations, Belsley et al. [3] compared the influences of these two observations in (1.2) on an ordinary least squares estimator (OLSE) by utilization of the determinant ratio of covariance matrices of estimators.

In this section, we present an application of the above results to compare the effects of observations (1.2) on BLUE under the model with n + 2 observations according to the covariance matrix criterion without the restrictions (3.1) on matrix X ^ j , j = 1 , 2 . With m 1 = m 2 = 1 , we write (1.2) as

(3.2) r j = α j ′ β + e j , E ( e j ) = 0 , Cov ( e j ) = σ j 2 > 0 , j = 1 , 2 .

By Corollary 2.2, the following results can be given.

Corollary 3.1

Under Assumptions 2.1 and 2.3, and let l 1 = l 2 = m 1 = m 2 = 1 in Assumption 2.3. Then,

(a) Cov ( BLUE ( X β ∣ M 1 ) ) ≼ Cov ( BLUE ( X β ∣ M 2 ) ) ⇔ α 2 ′ α 2 ≤ α 1 ′ α 1 .

(b) The inequality Cov ( BLUE ( X β ∣ M 1 ) ) ≺ Cov ( BLUE ( X β ∣ M 2 ) ) only hold under α 2 ′ α 2 < α 1 ′ α 1 and n = 1 .

As to σ 1 2 = 0 or σ 2 2 = 0 in (3.2), similar conclusions can also be drawn. We omit them here. If we would like to delete the observation r 1 from the model M 1 with m 1 = 1 , then considering the model M 1 after deleting the observation r 1 is equivalent to considering a new model

y r 1 = X β α 1 ′ β + e ^ n + 1 α + ε e 1 ,

where e ^ n + 1 = ( 0 , … , 0 , 1 ) ′ ∈ R ( n + 1 ) × 1 by using Theorem 2.1 because of the fact

C α 1 1 ∩ C X ′ 0 = { 0 } .

This has an important application to deletion diagnostics [37].

In order to explain the theoretical conclusions, we consider an example regarding piglet fattening due to Section 1.3 of [38]. This example can be given by a linear regression model

(3.3) y = X β + ε ,

where

(3.4) y = y 11 y 12 y 13 y 21 y 22 y 23 , X = 1 1 0 5.0 1 1 0 5.0 1 1 0 5.2 1 0 1 5.0 1 0 1 5.3 1 0 1 5.4 , β = μ β 1 β 2 γ , ε = ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 .

Set

(3.5) Cov ( ε ) = Σ = 1 1 1 0 0 0 1 1 1 0 0 0 1 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 .

These data originate from an experimental research on piglet fattening where several kinds of feeds are fed to various piglets. The experiment is usually implemented to compare the fattening effects of different feeds in terms of the amount of weight gained. Here, y i j is the number of weight obtained by the jth piglet fed to the ith feed, i = 1 , 2, j = 1 , 2, 3, μ means a total effect included in all the observations, β i denotes the effect of the i th feed, i = 1 , 2, γ represents the regression coefficient. Put

(3.6) M j = y 1 r j , X 1 β α j ′ β , Σ 1 0 0 1 , j = 1 , 2 ,

where

y 1 = y 11 y 12 y 13 y 21 , X 1 = 1 1 0 5.0 1 1 0 5.0 1 1 0 5.2 1 0 1 5.0 , Σ 1 = 1 1 1 0 1 1 1 0 1 1 2 0 0 0 0 1 , α 1 = ( 1 , 0 , 1 , 5.3 ) ′

α 2 = ( 1 , 0 , 1 , 5.4 ) ′ , r 1 = y 22 and r 2 = y 23 .

Note that

C ( α 1 ) ⊆ C ( X 1 ′ ) , C ( α 2 ) ⊆ C ( X 1 ′ ) , C ( X 1 ) ⊆ C ( Σ 1 ) ,

that is, Assumptions 2.1–2.3 in Section 2 hold. Hence, the conditions in Corollary 3.1 are satisfied. Since

α 1 ′ α 1 = 1 2 + 0 2 + 1 2 + 5 . 3 2 < 1 2 + 0 2 + 1 2 + 5 . 4 2 = α 2 ′ α 2 ,

so we obtain

Cov ( BLUE ( X 1 β ∣ M 2 ) ) ≺ Cov ( BLUE ( X 1 β ∣ M 1 ) )

in view of Corollary 3.1, i.e., the observation y 22 has a bigger effect than the observation y 23 on the BLUE according to the covariance matrix criterion. Trivially,

X 1 α j ′ ′ X 1 α j ′ ≻ 0 , j = 1 , 2 ,

does not hold, which leads to the ineffectiveness of many ways to compare the impact of observations on the BLUE, such as Cook’s distance, Welsch-Kuh’s distance, Andrews-Pregibon statistic, and so on.

4 Conclusion

The author made a comparison between covariance matrices of BLUEs under M and M 1 in (1.3), which showed that stochastic linear restrictions could improve the corresponding BLUE with respect to the covariance matrix criterion. Based on the results derived in Theorem 2.1, we gained some necessary and sufficient conditions for a variety of equalities and inequalities of covariance matrices of BLUEs under M 1 and M 2 in (1.3) to hold. The comparison findings clearly give us some answers on how to evaluate the effects of different stochastic linear restrictions on BLUE. It is worth noting that if V 1 , V 2 , and Σ are unknown, then Theorems 2.1 and 2.2 have no use by reason of involvement of V 1 , V 2 , and Σ . In this situation, they can be substituted for the corresponding estimators, and these are other kinds of work that are not discussed in this article. The OLSE defined by the optimality criterion different from BLUE is frequently employed in the literature due to its excellent properties. Therefore, under Assumption 2.1, it will be interesting to investigate how to make a comparison between the covariance matrices of OLSEs under M 1 and M 2 in (1.3). In addition, how to compare the covariance matrices of OLSEs and BLUEs under M 1 and M 2 in (1.3) is also a challenging work.

Acknowledgments

The author would like to thank the handling editor and anonymous reviewers for their insightful comments and suggestions.

Funding information: This research was supported by the Natural Science Foundation of Anhui Provincial Education Department (2024AH050299), and the key discipline construction project of Anhui Science and Technology University (XK-XJGY002).
Author contribution: The author has accepted responsibility for the whole content of the manuscript and approved its submission.
Conflict of interest: The author declares no conflict of interest.
Ethical approval: Not applicable.
Data availability statement: Not applicable.

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Received: 2024-10-30

Revised: 2025-04-12

Accepted: 2025-05-26

Published Online: 2025-07-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2025-0152

Keywords for this article

weakly singular linear regression model; restricted model; superiority; BLUE

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