A matrix approach to determine optimal predictors in a constrained linear mixed model

Nesrin Güler; Melek Eriş Büyükkaya

doi:10.1515/math-2024-0114

Article Open Access

A matrix approach to determine optimal predictors in a constrained linear mixed model

Nesrin Güler and Melek Eriş Büyükkaya

Published/Copyright: December 26, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

For a general vector of all unknown vectors in a constrained linear mixed model (CLMM), this study compared the dispersion matrices of the best linear unbiased predictors with any symmetric matrix for determining the optimality of predictors among others. Using the methodology of block matrix inertias and ranks as well as elementary block matrix operations, some equalities and inequalities are derived for comparisons. Additionally, the comparison findings for the CLMM reduce to special instances of the general vector of all unknown vectors. We also provide some comparison findings between the CLMM and its unconstrained form. Finally, we give a numerical example to illustrate our theoretical findings.

Keywords: best linear unbiased predictor; comparison; constrained linear mixed model; inertia; rank

MSC 2010: 15A03; 62H12; 62J05

1 Introduction

Linear mixed models ( LMM s), which allow the use of both fixed and random effects in the same analysis, are a type of linear regression model. These models ensure practical and useful approaches for statistical inferences by considering the variability of model parameters influenced by numerous factors affecting the response variable. For the analysis of longitudinal and correlated data, which can be encountered in many different fields as well as in the field of statistics, LMM s and some of their special versions such as hierarchical, nested, multilevel, and linear random effects models are frequently utilized.

In statistical research, there may exist relevant extraneous information in the form of exact or stochastic restrictions on unknown parameters in linear models. Classical restrictions, mixed estimation restrictions, adding-up restrictions, and stochastic response restrictions are some examples of such restrictions [1]. If relevant extraneous information comes from theory or known knowledge of the relationships among unknown parameters, restrictions on unknown parameters, i.e., classical restrictions, arise, see, e.g., [2]. Usually, these restrictions are incorporated into the model assumptions. In such cases, by adding certain restrictions on unknown parameters to LMM assumptions, LMM s become constrained linear mixed models ( CLMM s). The reparameterization of models is one of the most used approaches for the adding process of exact linear restrictions to LMM s. Through this approach, a CLMM is made into a reparameterized LMM by setting the general solution of the linear restriction equation into the model, i.e., an unconstrained LMM for the new parameter vectors is derived from a CLMM by this approach, see, e.g., [3,4]. In the present study, we consider an LMM with an exact linear restriction on its parameters, i.e., we consider a CLMM and turn this model into its reparameterized form. We give the derivation to establish the best linear unbiased predictors ( BLUP s) of all unknown vectors in the models using some quadratic matrix optimization methods including matrix ranks and inertias together with some essential properties of the BLUP s.

The development of various potential predictors (or also potential estimators) of unknown parameter vectors in linear regression models and their specific situations is one of the main objectives of theoretical and practical studies. In order to define predictors of unknown parameter vectors in the model, various types of optimality criteria have been used in mathematics. The unbiasedness of predictors of unknown parameter vectors according to the parameter spaces in the model is one of the crucial characteristics among others. Finding the unbiased predictors with the smallest dispersion matrix among all the unbiased predictors makes sense when there are numerous unbiased predictors for the same parameter space. The BLUP by its definition corresponds to these two requirements. Thus, due to the minimizing property of the BLUP ’s dispersion matrix, the BLUP has a significant role to play in statistical inference theory and is frequently used as the foundation for evaluating the effectiveness of various predictors. Comparing any predictor and the BLUP for a unified form of all unknown parameters under the linear regression models in terms of the dispersion matrix of BLUP provides to determine the optimality of this compared predictor. The expressions that are obtained from the comparison issues of the predictors include various complicated matrix operations. Some of the methods to simplify expressions composed by matrices and their Moore-Penrose generalized inverses are based on conventional operations of matrices, as well as some known expressions for inverses or Moore-Penrose generalized inverses of matrices. The effectiveness of these methods is quite limited, and the processes involved are quite tedious. Statisticians, in general, have widely used the Löwner partial ordering ( LPO ) for symmetric matrices to obtain certain satisfactory results in problems of comparing different predictors. On the other hand, in recent years, many new expansion formulas for calculating inertias and ranks of matrices have been established, and these formulas can be used to characterize various types of equalities and inequalities between dispersion matrices of BLUPs and other predictors. In other words, the methodology of block matrix inertias and ranks and also elementary block matrix operations ( EBMO s) is quite effective to remove complex matrix operations in various complicated matrix expressions or equalities, while the results obtained can easily be represented as some simple inertia and rank equalities.

In this study, we deal with the problems of comparing predictors for determining optimal predictors among others. We construct numerous equality and inequality relations among the dispersion matrix of the BLUP and any symmetric matrix. The main purpose of this study is to describe how to establish exact formulas for the comparisons of any predictor and the BLUP for the same general vector of all unknown vectors under CLMM by using the methodology of block matrix inertias and ranks as well as EBMO s. In this way, this article offers new theoretical techniques and innovative approaches in the comparison of two predictors, one of which is BLUP , by establishing various equalities and inequalities CLMM by using the matrix inertia and rank methodology under the general assumptions. As mentioned earlier, the reason for using the dispersion matrix of BLUP in comparisons as a comparison criterion to obtain optimal predictors is based on the BLUP definition, which requires a minimum covariance matrix of unbiased predictors of unknown vectors. In addition, the reason for preferring the methodology of block matrix inertias and ranks with EBMO s is to convert the complex matrix expressions into simpler forms since the operations we encounter within the framework of the subject considered in the study include complex matrix expressions. Thus, what we want to say is that the approaches used in the study of the problem of comparing any predictor and the BLUP for determining optimality of predictors among others reduce to the problem of comparing quantities since the inertia and rank of a matrix are simple quantities to understand and calculate.

In order to calculate the set of formulas based on the inertias of differences between any symmetric matrix and the dispersion matrix of the BLUP of all unknown vectors in the CLMM , first, a CLMM with its reparameterized form is introduced, and then, the classical concepts of the predictability, estimability, and some fundamental results on the BLUP are reviewed. After constructing the comparison formulas, as an application, the results obtained are reduced to the corresponding results under the restricted and unrestricted model, and finally, a numerical example is given to illustrate the theoretical results depending on whether the dispersion matrix is non-singular or singular. LMM s have been widely studied in the statistical literature from various aspects. We may refer to the following studies for LMM s with restrictions: [5–7], for linear random effects models with restrictions: [8,9], and for linear models with restrictions: [10–14], among others. The comparison of dispersion matrices under different types of models with or without restrictions has been presented, e.g., [15–17].

We first describe the notations that are utilized in this work before moving on, and then, we present a number of formulas for calculating the ranks and inertias of block matrices to characterize equalities and inequalities of matrices in the following two lemmas, see, [18]. R k × t is written to represent the set of all k × t real matrices. ( ⋅ ) ′ denotes the transpose of a matrix; furthermore, r ( ⋅ ) is used for the rank of a matrix, and for the column space of a matrix, we use C ( ⋅ ) . Λ ⊥ = I k − Λ Λ + and F Λ = I t − Λ + Λ represent the orthogonal projectors related to Λ ∈ R k × t , where Λ + is the Moore-Penrose generalized inverse of Λ and I k ∈ R k × k and I t ∈ R t × t are the identity matrices. i + ( ⋅ ) and i − ( ⋅ ) represent the positive and negative inertias of a symmetric matrix, while i ± ( ⋅ ) and i ∓ ( ⋅ ) are used to denote both signs of inertias jointly. When comparing dispersion matrices, the notations ≼ and ≽ are employed, namely, for two symmetric matrices Λ 1 and Λ 2 of the same size, the inequalities Λ 1 − Λ 2 ≼ 0 ( Λ 1 ≼ Λ 2 ) and Λ 1 − Λ 2 ≽ 0 ( Λ 1 ≽ Λ 2 ) mean that the differences Λ 1 − Λ 2 are negative semi-definite and positive semi-definite (psd) matrices in the LPO , respectively. In addition, E ( ⋅ ) is used for the expectation of a random vector, and we also use D ( ⋅ ) and cov ( ⋅ , ⋅ ) to denote the dispersion and covariance matrices of random vectors, respectively.

Lemma 1.1

Let Λ , ϒ ∈ R k × t , or, let Λ = Λ ′ , ϒ = ϒ ′ ∈ R k × k . Then,

(1) r ( Λ − ϒ ) = 0 ⇔ Λ = ϒ ,

(2) i − ( Λ − ϒ ) = 0 ⇔ Λ ≽ ϒ a n d i + ( Λ − ϒ ) = 0 ⇔ Λ ≼ ϒ .

Lemma 1.2

Let Λ = Λ ′ ∈ R k × k , Γ = Γ ′ ∈ R t × t , ϒ ∈ R k × t , and l ∈ R . Then,

(3) r ( Λ ) = i + ( Λ ) + i − ( Λ ) ,

(4) i ± ( l Λ ) = i ± ( Λ ) , i f l > 0 , a n d i ± ( l Λ ) = i ∓ ( Λ ) , i f l < 0 ,

(5) i ± Λ ϒ ϒ ′ Γ = i ± Λ − ϒ − ϒ ′ Γ = i ∓ − Λ ϒ ϒ ′ − Γ ,

(6) i ± Λ 0 0 Γ = i ± ( Λ ) + i ± ( Γ ) a n d i + 0 Λ Λ ′ 0 = i − 0 Λ Λ ′ 0 = r ( Λ ) ,

(7) i ± Λ ϒ ϒ ′ 0 = r ( ϒ ) + i ± ( ϒ ⊥ Λ ϒ ⊥ ) ,

(8) i + Λ ϒ ϒ ′ 0 = r Λ , ϒ a n d i − Λ ϒ ϒ ′ 0 = r ( ϒ ) , i f Λ ≽ 0 ,

(9) i ± Λ ϒ ϒ ′ Γ = i ± ( Λ ) + i ± ( Γ − ϒ ′ Λ + ϒ ) , i f C ( ϒ ) ⊆ C ( Λ ) .

2 CLMM with its reparameterized form and BLUPs

This section of this article is devoted to the introduction of a reparameterized model setup, BLUP s under this reparameterized model, and the corresponding results.

Let us consider a CLMM as follows:

(10) C : y = X α + Z γ + ε , C α = c , with E γ ε = 0 , and D γ ε = cov γ ε , γ ε = D ( γ ) cov ( γ , ε ) cov ( ε , γ ) D ( ε ) = σ 2 V 1 V 2 V 2 ′ V 3 ≔ σ 2 V ,

with the following general vector to consider the simultaneous prediction problem of all unknown parameters

(11) ψ = P α + R γ + S ε = P α + R , S γ ε ,

where y ∈ R n × 1 is a vector of responses, X ∈ R n × k , Z ∈ R n × p , and C ∈ R m × k are the known matrices of arbitrary ranks, α ∈ R k × 1 is a parameter vector of fixed effects, γ ∈ R p × 1 is a vector of random effects, ε ∈ R n × 1 is a vector of random error, c ∈ R m × 1 is a known vector, V ∈ R ( n + p ) × ( n + p ) is a known psd matrix of arbitrary rank, σ 2 is a positive unknown parameter, and P ∈ R s × k , R ∈ R s × p , and S ∈ R s × n are given matrices of arbitrary ranks. Furthermore, C α = c is supposed to be a consistent linear matrix equation. According to assumptions in (10),

D ( y ) = σ 2 Z , I n V Z , I n ′ ≔ σ 2 Λ V Λ ′ ,

D ( ψ ) = σ 2 R , S V R , S ≔ σ 2 Γ V Γ ′ , and cov ( ψ , y ) = σ 2 Γ V Λ ′

are obtained, where Λ = Z , I n and Γ = R , S .

To draw any statistical conclusions from (10), first, it is needed to convert (10) into an implicitly CLMM . For doing this, we substitute the vector α = C + c + F C u , the general solution of the matrix equation C α = c , into the model equation in (10), where u ∈ R k × 1 is an arbitrary vector that has been reparameterized, and thereby, this situation yields the following reparameterized LMM :

(12) ℛ : r = X F C u + Z γ + ε ,

where r = y − X C + c ; thus, predictions under C can be derived from ℛ . Correspondingly, ψ in (11) becomes the following reparameterized vector:

(13) ϕ = P F C u + R γ + S ε = P F C u + R , S γ ε ,

where ϕ = ψ − P C + c . In the study, we suppose that the models C and ℛ are consistent, i.e., y ∈ C X , Λ V Λ ′ and r ∈ C X F C , Λ V Λ ′ hold with probability 1, respectively, see, e.g., [19]. According to these consistency requirements, clearly, if ℛ is consistent, then C is consistent.

The definitions of predictability and the BLUP s under the models are given as follows, see, e.g., [20–22].

Definition 2.1

Let C , ψ , ℛ , and ϕ be as given in (10)–(13), respectively.

ψ is said to be predictable under C if there exist K ∈ R s × n and k ∈ R s × 1 such that E ( K y + k − ψ ) = 0 holds, i.e., C ( P ′ ) ⊆ C X ′ , C ′ , in this case, P α is also estimable under C .
Let ψ be predictable under C . D ( K y + k − ψ ) = min s.t. E ( K y + k − ψ ) = 0 holds in the LPO , and K y + k is defined to be the BLUP of ψ under C and is denoted by
K y + k = BLUP C ( ψ ) = BLUP C ( P α + R γ + S ε ) .
Moreover, K y + k corresponds to the best linear unbiased estimator ( BLUE ) of P α , denoted by BLUE C ( P α ) , by writing R = 0 and S = 0 in ψ .
ϕ is said to be predictable under ℛ if there exists K ∈ R s × n such that E ( K r − ϕ ) = 0 holds, i.e., C ( F C P ′ ) ⊆ C ( F C X ′ ) ; in this case, P F C u is also estimable under ℛ .
Let ϕ be predictable under ℛ . D ( K r − ϕ ) = min s.t. E ( K r − ϕ ) = 0 holds in the LPO , and K r is defined to be the BLUP of ϕ under ℛ and is written as
K r = BLUP ℛ ( ϕ ) = BLUP ℛ ( P F C u + R γ + S ε ) .
Moreover, K r = BLUE ℛ ( P F C u ) when R = 0 and S = 0 in ϕ .

Note that ψ is predictable under C ⇔ ϕ is predictable under ℛ , see, [23, Theorem 2.4]. It is easily seen from Definition 2.1 that γ , ε , R γ , S ε , R γ + S ε , and X F C u + Z γ + ε are always predictable and X F C u is always estimable under ℛ . Furthermore, u is estimable under ℛ ⇔ r ( X F C ) = k . It is also mentioned that

D ( K y + k − ψ ) = min s.t. E ( K y + k − ψ ) = 0 ⇔ D ( K r − ϕ ) = min s.t. E ( K r − ϕ ) = 0 ,

i.e., the following equality

(14) BLUP C ( ψ ) = P C + c + BLUP ℛ ( ϕ )

holds, see [23], and thereby,

(15) ψ − BLUP C ( ψ ) = ψ − P C + c − BLUP ℛ ( ϕ ) , i.e., D [ ψ − BLUP C ( ψ ) ] = D [ ϕ − BLUP ℛ ( ϕ ) ] .

It is clearly seen from (14) that the BLUP of ψ under C is obtained from the BLUP of ϕ under ℛ ; in other words, the BLUP of all unknown vectors under CLMM s can be derived from some well-known standard procedures that have been derived for unconstrained LMM s. It is also seen from (15) that the comparison problem of dispersion matrix of BLUP of ψ under C corresponds to the comparison problem of dispersion matrix of BLUP of ϕ under ℛ .

Concerning the existence and representations of BLUP s of unknown vectors ψ and ϕ under the models, as well as some of the properties of these predictors are presented below, see also [24], and for different approaches, see, e.g., [19,22].

Lemma 2.1

Consider the vectors ψ and ϕ that are given in (11) and (13), respectively. Suppose that ϕ is predictable under ℛ . Consider the vectors K r and T r that are unbiased linear predictors for ϕ . Then,

(16) max E ( T r − ϕ ) = 0 i + ( D ( K r − ϕ ) − D ( T r − ϕ ) ) = r K , − I s Λ V Λ ′ Λ V Γ ′ Γ V Λ ′ Γ V Γ ′ X F C P F C ⊥

is the maximal positive inertia of D ( K r − ϕ ) − D ( T r − ϕ ) s.t. T X F C = P F C . Hence,

(17) D ( K r − ϕ ) = min s.t. E ( K r − ϕ ) = 0 ⇔ K r = BLUP ℛ ( ϕ ) ⇔ K X F C , Λ V Λ ′ ( X F C ) ⊥ = P F C , Γ V Λ ′ ( X F C ) ⊥ .

This matrix equation is consistent. According to the general solution of the equation,

(18) BLUP ℛ ( ϕ ) = K r = P F C , Γ V Λ ′ ( X F C ) ⊥ J r + + U J r ⊥ r

is written, where J r = X F C , Λ V Λ ′ ( X F C ) ⊥ and U ∈ R s × n is an arbitrary matrix. In particular, K is unique ⇔ r ( J r ) = n and BLUP ℛ ( ϕ ) is unique ⇔ ℛ is consistent. Additionally,

(19) r ( J r ) = r X F C , Λ V Λ ′ = r X F C , ( X F C ) ⊥ Λ V Λ ′ , C ( J r ) = C X F C , Λ V Λ ′ = C X F C , ( X F C ) ⊥ Λ V Λ ′ .

Correspondingly, the BLUP of ψ under C can be written as

(20) BLUP C ( ψ ) = K y + k = P C + c + P F C , Γ V Λ ′ ( X F C ) ⊥ J r + + U J r ⊥ r ,

and the BLUE of P F C u under C can be written as

(21) BLUE C ( P F C u ) = K y + k = P C + c + P F C , 0 J r + + U J r ⊥ r .

Lemma 2.2

Consider the vectors ψ and ϕ that are given in (11) and (13), respectively. Suppose that ϕ is predictable under ℛ . Then,

D [ BLUP ℛ ( ϕ ) ] = σ 2 D r J r + Λ V Λ ′ ( J r + ) ′ D r ′ ,

cov { BLUP ℛ ( ϕ ) , ϕ } = σ 2 D r J r + Λ V Γ ′ ,

(22) D [ ϕ − BLUP ℛ ( ϕ ) ] = D [ ψ − BLUP C ( ψ ) ] = σ 2 ( D r J r + Λ − Γ ) V ( D r J r + Λ − Γ ) ′ ,

where D r = P F C , Γ V Λ ′ ( X F C ) ⊥ and J r = X F C , Λ V Λ ′ ( X F C ) ⊥ . Also, when R = 0 and S = 0 , (22) corresponds to

(23) D [ BLUE ℛ ( P F C u ) ] = D [ BLUE C ( P α ) ] = σ 2 P F C , 0 J r + Λ V P F C , 0 J r + Λ ′ .

3 Characterizing relationships of BLUPs’ dispersion matrices under a CLMM

In this section, various comparisons of predictors under C are made using ℛ , and conclusions about special cases are drawn using both rank and inertia formulas for block matrices with EBMO s.

Theorem 3.1

Let C , ψ , ℛ , and ϕ be as given in (10)–(13), respectively. Suppose that ϕ is predictable under ℛ . Let Q ∈ R s × s be any symmetric matrix and denote

G = Λ V Λ ′ Λ V Γ ′ X F C Γ V Λ ′ Γ V Γ ′ − σ − 2 Q P F C F C X ′ F C P ′ 0 = σ − 2 D ( y ) σ − 2 cov ( y , ϕ ) X F C σ − 2 cov ( ϕ , y ) σ − 2 D ( ϕ ) − σ − 2 Q P F C F C X ′ F C P ′ 0 .

Then,

(24) i + ( Q − D [ ϕ − BLUP ℛ ( ϕ ) ] ) = i − ( G ) − r ( X F C ) ,

(25) i − ( Q − D [ ϕ − BLUP ℛ ( ϕ ) ] ) = i + ( G ) − r X F C , Λ V Λ ′ ,

(26) r ( Q − D [ ϕ − BLUP ℛ ( ϕ ) ] ) = r ( G ) − r ( X F C ) − r X F C , Λ V Λ ′ .

As a result, the outcomes listed below are provided.

Q ≽ D [ ϕ − BLUP ℛ ( ϕ ) ] ⇔ Q ≽ D [ ψ − BLUP C ( ψ ) ] ⇔ i + ( G ) = r X F C , Λ V Λ ′ ,
Q ≼ D [ ϕ − BLUP ℛ ( ϕ ) ] ⇔ Q ≼ D [ ψ − BLUP C ( ψ ) ] ⇔ i − ( G ) = r ( X F C ) ,
Q = D [ ϕ − BLUP ℛ ( ϕ ) ] ⇔ Q = D [ ψ − BLUP C ( ψ ) ] ⇔ r ( G ) = r X F C , Λ V Λ ′ + r ( X F C ) .

Proof

Let us consider D [ ϕ − BLUP ℛ ( ϕ ) ] in (22) and apply (9) to the difference between a symmetric matrix Q and D [ ϕ − BLUP ℛ ( ϕ ) ] . Then, we obtain

(27) i ± ( Q − D [ ϕ − BLUP ℛ ( ϕ ) ] ) = i ± ( Q − σ 2 ( D r J r + Λ − Γ ) V ( D r J r + Λ − Γ ) ′ ) = i ± V V Λ ′ ( J r + ) ′ D r ′ − V Γ ′ D r J r + Λ V − Γ V σ − 2 Q − i ± ( V ) = i ± V − V Γ ′ − Γ V σ − 2 Q + V Λ ′ 0 0 D r 0 J r J r ′ 0 + Λ V 0 0 D r ′ − i ± ( V ) ,

where D r = P F C , Γ V Λ ′ ( X F C ) ⊥ and J r = X F C , Λ V Λ ′ ( X F C ) ⊥ . Note that the following column space inclusions hold:

C ( Λ V ) = C ( Λ V Λ ′ ) ⊆ C X F C , Λ V Λ ′ = C ( J r ) (by (19)),
C ( F C P ′ ) ⊆ C ( F C X ′ ) (from Definition 2.1) and C ( Λ V Γ ′ ) ⊆ C ( Λ V ) = C ( Λ V Λ ′ ) , and thereby, C ( D r ′ ) ⊆ C ( J r ′ ) .

In this case, reapplying (9) to (27) and also using (4) and (6) with the EBMO s, the expression in (27) becomes

(28) i ± 0 − J r Λ V 0 − J r ′ 0 0 D r ′ V Λ ′ 0 V − V Γ ′ 0 D r − Γ V σ − 2 Q − i ∓ 0 J r J r ′ 0 − i ± ( V ) = i ± − Λ V Λ ′ − J r Λ V Γ ′ − J r ′ 0 D r ′ Γ V Λ ′ D r σ − 2 Q − Γ V Γ ′ − r ( J r ) .

By setting D r and J r into (28) and also using (5)–(7) and (19) with the EBMO s,

i ± − Λ V Λ ′ − X F C − Λ V Λ ′ ( X F C ) ⊥ Λ V Γ ′ − F C X ′ 0 0 F C P ′ − ( X F C ) ⊥ Λ V Λ ′ 0 0 ( X F C ) ⊥ Λ V Γ ′ Γ V Λ ′ P F C Γ V Λ ′ ( X F C ) ⊥ σ − 2 Q − Γ V Γ ′ − r X F C , Λ V Λ ′

(29) = i ± − Λ V Λ ′ − X F C Λ V Γ ′ − F C X ′ 0 F C P ′ Γ V Λ ′ P F C σ − 2 Q − Γ V Γ ′ + i ± ( ( X F C ) ⊥ Λ V Λ ′ ( X F C ) ⊥ ) − r X F C , Λ V Λ ′ = i ∓ Λ V Λ ′ Λ V Γ ′ X F C Γ V Λ ′ Γ V Γ ′ − σ − 2 Q P F C F C X ′ F C P ′ 0 + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′

is obtained. From (8),

i + Λ V Λ ′ X F C F C X ′ 0 = r X F C , Λ V Λ ′ and i − Λ V Λ ′ X F C F C X ′ 0 = r ( X F C )

are written, and then, (24) and (25) are obtained from (29). (3) states that (26) is the result of summing the equalities in (24) and (25). Lemma 1.1 applied to (24)–(26) gives items (1)–(3).□

Corollary 3.1

Let C and ℛ be as given in (10) and (12), respectively, and denote

G 1 = σ − 2 D ( y ) 0 X F C 0 − σ − 2 Q X F C F C X ′ F C X ′ 0 , G 2 = σ − 2 D ( y ) 0 X F C 0 − σ − 2 Q I k F C X ′ I k 0 , G 3 = σ − 2 D ( y ) σ − 2 [ Z D ( γ ) + cov ( ε , γ ) ] X F C σ − 2 [ D ( γ ) Z ′ + cov ( γ , ε ) ] σ − 2 [ D ( γ ) − Q ] 0 F C X ′ 0 0 , G 4 = σ − 2 D ( y ) σ − 2 [ D ( ε ) + Z cov ( γ , ε ) ] X F C σ − 2 [ D ( ε ) + cov ( ε , γ ) Z ′ ] σ − 2 [ D ( ε ) − Q ] 0 F C X ′ 0 0 .

X F C u is always estimable under ℛ , and then,
1. Q ≽ D [ BLUE ℛ ( X F C u ) ] ⇔ Q ≽ D [ BLUE C ( X α ) ] ⇔ i + ( G 1 ) = r X F C , Λ V Λ ′ ,
2. Q ≼ D [ BLUE ℛ ( X F C u ) ] ⇔ Q ≼ D [ BLUE C ( X α ) ] ⇔ i − ( G 1 ) = r ( X F C ) ,
3. Q = D [ BLUE ℛ ( X F C u ) ] ⇔ Q = D [ BLUE C ( X α ) ] ⇔ r ( G 1 ) = r X F C , Λ V Λ ′ + r ( X F C ) .
Suppose that u is estimable under ℛ , i.e., r ( X F C ) = k holds. Then,
1. Q ≽ D [ BLUE ℛ ( u ) ] ⇔ Q ≽ D [ BLUE C ( α ) ] ⇔ i + ( G 2 ) = r X F C , Λ V Λ ′ ,
2. Q ≼ D [ BLUE ℛ ( u ) ] ⇔ Q ≼ D [ BLUE C ( α ) ] ⇔ i − ( G 2 ) = k ,
3. Q = D [ BLUE ℛ ( u ) ] ⇔ Q = D [ BLUE C ( α ) ] ⇔ r ( G 2 ) = r X F C , Λ V Λ ′ + k .
γ and ε are always predictable under ℛ , and then,
1. Q ≽ D [ γ − BLUP ℛ ( γ ) ] ⇔ Q ≽ D [ γ − BLUP C ( γ ) ] ⇔ i + ( G 3 ) = r X F C , Λ V Λ ′ ,
2. Q ≼ D [ γ − BLUP ℛ ( γ ) ] ⇔ Q ≼ D [ γ − BLUP C ( γ ) ] ⇔ i − ( G 3 ) = r ( X F C ) ,
3. Q = D [ γ − BLUP ℛ ( γ ) ] ⇔ Q = D [ γ − BLUP C ( γ ) ] ⇔ r ( G 3 ) = r X F C , Λ V Λ ′ + r ( X F C ) ,
4. Q ≽ D [ ε − BLUP ℛ ( ε ) ] ⇔ Q ≽ D [ ε − BLUP C ( ε ) ] ⇔ i + ( G 4 ) = r X F C , Λ V Λ ′ ,
5. Q ≼ D [ ε − BLUP ℛ ( ε ) ] ⇔ Q ≼ D [ ε − BLUP C ( ε ) ] ⇔ i − ( G 4 ) = r ( X F C ) ,
6. Q = D [ ε − BLUP ℛ ( ε ) ] ⇔ Q = D [ ε − BLUP C ( ε ) ] ⇔ r ( G 4 ) = r X F C , Λ V Λ ′ + r ( X F C ) .

4 Application to a CLMM and its unconstrained form

Various new results can be derived by choosing special matrices instead of the matrix Q in Theorem 3.1. The results of comparing the dispersion matrices of BLUP s under a CLMM and its unconstrained form are presented in this section.

Let us consider the following unconstrained LMM , obtained by considering the model C in (10) without restriction on parameter vector α ,

(30) ℳ : y = X α + Z γ + ε ,

with the same assumptions in (10). It is obvious that

(31) ψ is predictable under ℳ ⇔ C ( P ′ ) ⊆ C ( X ′ ) .

In this case, ψ is predictable under C .

Suppose that ψ is predictable under ℳ . There exist K ∈ R s × n such that

D ( K y − ψ ) = min s.t. E ( K y − ψ ) = 0 ⇔ K y = BLUP ℳ ( ψ ) ⇔ K X , Λ V Λ ′ X ⊥ = P , Γ V Λ ′ X ⊥ ,

and then, BLUP ℳ ( ψ ) = K y = ( [ P , Γ V Λ ′ X ⊥ ] J m + + U J m ⊥ ) y , where U ∈ R s × n is an arbitrary matrix and, in this case,

(32) D [ ψ − BLUP ℳ ( ψ ) ] = σ 2 ( D m J m + Λ − Γ ) V ( D m J m + Λ − Γ ) ′ ,

where D m = [ P , Γ V Λ ′ X ⊥ ] and J m = [ X , Λ V Λ ′ X ⊥ ] . Furthermore,

(33) r ( J m ) = r X , Λ V Λ ′ = r X , X ⊥ Λ V Λ ′ , C ( J m ) = C X , Λ V Λ ′ = C X , X ⊥ Λ V Λ ′ .

Also, when R = 0 and S = 0 , (32) corresponds to

(34) D [ BLUE ℳ ( P α ) ] = σ 2 P , 0 J m + Λ V P , 0 J m + Λ ′ .

Theorem 4.1

Let C , ψ , and ℳ be as given in (10), (11), and (30), respectively, and suppose that ψ is predictable under ℳ . Denote

H = − Λ V Λ ′ Λ V Λ ′ Λ V Γ ′ 0 X F C Λ V Λ ′ 0 0 X 0 Γ V Λ ′ 0 0 P 0 0 X ′ P ′ 0 0 F C X ′ 0 0 0 0 = − σ − 2 D ( y ) σ − 2 D ( y ) σ − 2 cov ( y , ψ ) 0 X F C σ − 2 D ( y ) 0 0 X 0 σ − 2 cov ( ψ , y ) 0 0 P 0 0 X ′ P ′ 0 0 F C X ′ 0 0 0 0 .

Then,

(35) i + ( D [ ψ − BLUP ℳ ( ψ ) ] − D [ ψ − BLUP C ( ψ ) ] ) = i + ( H ) − r X , Λ V Λ ′ − r ( X F C ) ,

(36) i − ( D [ ψ − BLUP ℳ ( ψ ) ] − D [ ψ − BLUP C ( ψ ) ] ) = i − ( H ) − r X F C , Λ V Λ ′ − r ( X ) ,

(37) r ( D [ ψ − BLUP ℳ ( ψ ) ] − D [ ψ − BLUP C ( ψ ) ] ) = r ( H ) − r X , Λ V Λ ′ − r ( X F C ) − r X F C , Λ V Λ ′ − r ( X ) .

As a result, the outcomes listed below are provided:

D [ ψ − BLUP ℳ ( ψ ) ] ≽ D [ ψ − BLUP C ( ψ ) ] ⇔ i − ( H ) = r X F C , Λ V Λ ′ + r ( X ) ,
D [ ψ − BLUP ℳ ( ψ ) ] ≼ D [ ψ − BLUP C ( ψ ) ] ⇔ i + ( H ) = r X , Λ V Λ ′ + r ( X F C ) ,
D [ ψ − BLUP ℳ ( ψ ) ] = D [ ψ − BLUP C ( ψ ) ] ⇔ r ( H ) = r X , Λ V Λ ′ + r ( X F C ) + r X F C , Λ V Λ ′ + r ( X ) .

Proof

By considering D [ ψ − BLUP ℳ ( ψ ) ] instead of the matrix Q in (29),

(38) i ± ( D [ ψ − BLUP ℳ ( ψ ) ] − D [ ψ − BLUP C ( ψ ) ] ) = i ∓ Λ V Λ ′ Λ V Γ ′ X F C Γ V Λ ′ Γ V Γ ′ P F C F C X ′ F C P ′ 0 − σ − 2 0 I s 0 D [ ψ − BLUP ℳ ( ψ ) ] 0 I s 0 ′ + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′

is obtained. Substituting D [ ψ − BLUP ℳ ( ψ ) ] in (32) into (38) and then applying (9) to (38), the expression in (38) becomes

(39) i ∓ V 0 − V Γ ′ 0 0 Λ V Λ ′ Λ V Γ ′ X F C − Γ V Γ V Λ ′ Γ V Γ ′ P F C 0 F C X ′ F C P ′ 0 + V Λ ′ 0 0 0 0 D m 0 0 0 J m J m ′ 0 + Λ V 0 0 0 0 0 D m ′ 0 − i ∓ ( V ) + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′ ,

where D m = P , Γ V Λ ′ X ⊥ and J m = X , Λ V Λ ′ X ⊥ . Note that the following column space inclusions C ( Λ V ) = C ( Λ V Λ ′ ) ⊆ C X , Λ V Λ ′ = C ( J m ) (by (33)), C ( P ′ ) ⊆ C ( X ′ ) (by (31)) and C ( Λ V Γ ′ ) ⊆ C ( Λ V ) = C ( Λ V Λ ′ ) , and thereby, C ( D m ′ ) ⊆ C ( J m ′ ) hold. In this case, reapplying (9) to (39) and also using (4) and (6) with the EBMO s, the expression in (39) is equivalently written as follows:

(40) i ∓ 0 − J m Λ V 0 0 0 − J m ′ 0 0 0 D m ′ 0 V Λ ′ 0 V 0 − V Γ ′ 0 0 0 0 Λ V Λ ′ Λ V Γ ′ X F C 0 D m − Γ V Γ V Λ ′ Γ V Γ ′ P F C 0 0 0 F C X ′ F C P ′ 0 − i ± 0 J m J m ′ 0 − i ∓ ( V ) + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′ .

By setting D m and J m into (40) and also using (5)–(7) and (33) with the EBMO s, (40) is equivalently written as

i ∓ 0 − X − Λ V Λ ′ X ⊥ Λ V 0 0 0 − X ′ 0 0 0 0 P ′ 0 − X ⊥ Λ V Λ ′ 0 0 0 0 X ⊥ Λ V Γ ′ 0 V Λ ′ 0 0 V 0 − V Γ ′ 0 0 0 0 0 Λ V Λ ′ Λ V Γ ′ X F C 0 P Γ V Λ ′ X ⊥ − Γ V Γ V Λ ′ Γ V Γ ′ P F C 0 0 0 0 F C X ′ F C P ′ 0 − r X , Λ V Λ ′ − i ∓ ( V ) + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′ = i ∓ − Λ V Λ ′ − X − Λ V Λ ′ X ⊥ 0 Λ V Γ ′ 0 − X ′ 0 0 0 P ′ 0 − X ⊥ Λ V Λ ′ 0 0 0 X ⊥ Λ V Γ ′ 0 0 0 0 Λ V Λ ′ Λ V Γ ′ X F C Γ V Λ ′ P Γ V Λ ′ X ⊥ Γ V Λ ′ 0 P F C 0 0 0 F C X ′ F C P ′ 0 − r X , Λ V Λ ′ + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′

(41) = i ∓ − Λ V Λ ′ − X 0 Λ V Γ ′ 0 − X ′ 0 0 P ′ 0 0 0 Λ V Λ ′ Λ V Γ ′ X F C Γ V Λ ′ P Γ V Λ ′ 0 P F C 0 0 F C X ′ F C P ′ 0 + i ∓ ( X ⊥ Λ V Λ ′ X ⊥ ) − r X , Λ V Λ ′ + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′ = i ± − Λ V Λ ′ Λ V Λ ′ Λ V Γ ′ 0 X F C Λ V Λ ′ 0 0 X 0 Γ V Λ ′ 0 0 P 0 0 X ′ P ′ 0 0 F C X ′ 0 0 0 0 + i ∓ Λ V Λ ′ X X ′ 0 − r ( X ) − r X , Λ V Λ ′ + i ± Λ V Λ ′ X F C F C X ′ 0 − r ( X F C ) − r X F C , Λ V Λ ′ .

Using (8), (35) and (36) are obtained from (41). From (35) and (36), (37) is obtained. Lemma 1.1 applied to (35)–(37) gives items (1)–(3).□

For various selections of the matrices P , R , and S in ψ , many results can be drawn from Theorem 4.1. Specifically, when R = 0 , S = 0 , and P = X in ψ , Corollary 4.1 is obtained.

Corollary 4.1

Let C and ℳ be as given in (10) and (30), respectively, X α is always estimable under C and ℳ . Denote

H 1 = − Λ V Λ ′ Λ V Λ ′ 0 0 X F C Λ V Λ ′ 0 0 X 0 0 0 0 X 0 0 X ′ X ′ 0 0 F C X ′ 0 0 0 0 .

D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE C ( X α ) ] ⇔ i − ( H 1 ) = r X F C , Λ V Λ ′ + r ( X ) ,
D [ BLUE ℳ ( X α ) ] ≼ D [ BLUE C ( X α ) ] ⇔ i + ( H 1 ) = r X , Λ V Λ ′ + r ( X F C ) ,
D [ BLUE ℳ ( X α ) ] = D [ BLUE C ( X α ) ] ⇔ r ( H 1 ) = r X , Λ V Λ ′ + r ( X F C ) + r X F C , Λ V Λ ′ + r ( X ) .

5 Numerical example

In this section, we use a numerical example to explain the theoretical results presented in Sections 3 and 4. We focus on an example including real data used by [25]. The data have also been studied by many authors such as [26, p. 77] and [27]. The first lactation yields of dairy cows with herd effects and sire additive genetic merits are included in this data. The yields are assigned as the response ( y ), while herd effects are treated as fixed effects represented by h i , i = 1 , 2 , 3 , where h i is the environmental effect of the i th herd, and sire additive genetic merits are treated as random effects represented by s j , j = 1 , 2 , 3 , 4 , corresponding to sires A , B , C , and D , where s j is the effect of the j th sire on the lactation yield of the daughter. Therefore, we fit the model y = X α + Z γ + ε with

X = 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 , Z = 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 , α = h 1 h 2 h 3 , and γ = s 1 s 2 s 3 s 4 ,

to the data. Let us consider the restriction C α = c with C = 1 1 0 . This restriction may be a known fact from the theory or experiment view. The matrix V 1 is assumed to be 0.1 times the identity matrix and V 3 is taken to be the identity matrix as in [25].

As it is widely recognized, two frequently employed estimators within the framework of general linear models are the BLUE s and the ordinary least-squares estimators ( OLSE s). Both of these estimators exhibit a range of straightforward yet noteworthy properties. Given that these estimators are characterized by distinct optimality criteria, their expressions and properties are not necessarily the same. Therefore, it is only natural to explore potential connections between them. We use the aforementioned numerical example to illustrate the comparison results obtained in Section 3 by considering the BLUE and the OLSE as two estimators of X F c α under ℛ . Note that X F c α is estimable under ℛ . The relationship between the BLUE and the OLSE of X F c α , denoted as OLSE ℛ ( X F c α ) , under the model ℛ , is presented as follows. Using the NumPy library in Python and setting the aforementioned considerations and findings in the matrix G 1 in Corollary 3.1, we find the eigenvalues of the matrix G 1 as

− 3.074 , − 2.475 , − 1.100 , − 1.100 , 1.100 , 1.100 , 1.100 , 1.053 , 1.074 , 1.100 , 1.413 , 2.478 , 3.081 .

Thus, we can see that i + ( G 1 ) = 9 , i − ( G 1 ) = 4 , and thereby r ( G 1 ) = 13 . We also obtain r [ X F c , Λ V Λ ′ ] = 9 and r ( X F c ) = 2 . Therefore, i + ( G 1 ) = 9 = r [ X F c , Λ V Λ ′ ] ⇔ D [ BLUE ℛ ( X F c α ) ] ≼ D [ OLSE ℛ ( X F c α ) ] holds. This is already an expected result, meaning that this inequality is already a well-known property in statistical theory by definition of the estimators BLUE and OLSE . Since r ( X F c ) = 2 , i.e., i − ( G 1 ) ≠ r ( X F c ) , D [ OLSE ℛ ( X F c α ) ] ≼ D [ BLUE ℛ ( X F c α ) ] is not provided.

Now, we use the aforementioned numerical example again to illustrate the comparison results obtained in Section 4 by considering the BLUE s of X α under the models ℳ and C . Direct calculations show that

D [ BLUE ℳ ( X α ) ] = 0.55 0.55 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.55 0.55 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.03 0.03 0.38 0.38 0.38 0.03 0.03 0.03 0.03 0.03 0.03 0.38 0.38 0.38 0.03 0.03 0.03 0.03 0.03 0.03 0.38 0.38 0.38 0.03 0.03 0.03 0.03 0.02 0.02 0.03 0.03 0.03 0.30 0.30 0.30 0.30 0.02 0.02 0.03 0.03 0.03 0.30 0.30 0.30 0.30 0.02 0.02 0.03 0.03 0.03 0.30 0.30 0.30 0.30 0.02 0.02 0.03 0.03 0.03 0.30 0.30 0.30 0.30

and

D [ BLUE C ( X α ) ] = 0.21 0.21 − 0.21 − 0.21 − 0.21 − 0.01 − 0.01 − 0.01 − 0.01 0.21 0.21 − 0.21 − 0.21 − 0.21 − 0.01 − 0.01 − 0.01 − 0.01 − 0.21 − 0.21 0.21 0.21 0.21 0.01 0.01 0.01 0.01 − 0.21 − 0.21 0.21 0.21 0.21 0.01 0.01 0.01 0.01 − 0.21 − 0.21 0.21 0.21 0.21 0.01 0.01 0.01 0.01 − 0.01 − 0.01 0.01 0.01 0.01 0.29 0.29 0.29 0.29 − 0.01 − 0.01 0.01 0.01 0.01 0.29 0.29 0.29 0.29 − 0.01 − 0.01 0.01 0.01 0.01 0.29 0.29 0.29 0.29 − 0.01 − 0.01 0.01 0.01 0.01 0.29 0.29 0.29 0.29 .

Using the NumPy library in Python and setting the aforementioned considerations and findings in the matrix H 1 in Corollary 4.1, we find the eigenvalues of the matrix H 1 as

− 3.403 , − 2.883 , − 2.522 , − 2.385 , − 2.102 , − 1.881 , − 1.742 , − 1.702 , − 1.618 , − 1.618 , − 1.618 , − 1.326 ,

0.416 , 0.618 , 0.618 , 0.618 , 0.652 , 0.669 , 0.813 , 1.233 , 1.584 , 2.131 , 2.573 , 2.974 .

Thus, we can see that i + ( H 1 ) = 12 , i − ( H 1 ) = 12 , and thereby, r ( H 1 ) = 24 . We also obtain r [ X , Λ V Λ ′ ] = 9 and r ( X ) = 3 . Therefore, i − ( H 1 ) = 12 = r ( X ) + r [ X F c , Λ V Λ ′ ] , i.e., D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE C ( X α ) ] = D [ BLUE ℛ ( X F c α ) ] holds. Furthermore, D [ BLUE ℳ ( X α ) ] ≼ D [ BLUE C ( X α ) ] is not provided since i + ( H 1 ) ≠ r [ X , Λ V Λ ′ ] + r ( X F c ) . We also mention that since it is easily seen from the calculations that the difference D [ BLUE ℳ ( X α ) ] − D [ BLUE C ( X α ) ] or D [ BLUE ℳ ( X α ) ] − D [ BLUE ℛ ( X F c α ) ] is psd, we observe that the inequality D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE C ( X α ) ] or D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE ℛ ( X F c α ) ] already holds.

Finally, we add further calculations to illustrate the comparison results by assuming that the dispersion matrices in the aforementioned numerical example are singular. Consider the following singular matrices for the dispersion matrices V 1 and V 3 , respectively,

3 2 2 1 2 3 1 2 2 1 3 2 1 2 2 3 and 8 4 2 0 0 0 0 0 0 4 2 1 0 0 0 0 0 0 2 1 2 1 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 1 2

with σ 2 = 1 . Using the NumPy library in Python and setting the aforementioned considerations and findings in the matrix H 1 in Corollary 4.1, we find the eigenvalues of the matrix H 1 as

0.191 , 0.443 , 0.545 , 0.868 , 1.211 , 1.395 , 1.599 , 2.047 , 2.683 , 4.167 , 6.98 , 16.02

− 41.548 , − 17.719 , − 9.732 , − 5.67 , − 3.797 , − 2.363 , − 2.126 , − 1.916 , − 1.532 , − 1.324 , − 1.057 , − 0.366 .

Then, i + ( H 1 ) = i − ( H 1 ) = 12 , and thereby, r ( H 1 ) = 24 . Moreover, r [ X , Λ V Λ ′ ] = r [ X F c , Λ V Λ ′ ] = 9 , r ( X ) = 3 , and r ( X F c ) = 2 . Therefore, i − ( H 1 ) = 12 = r ( X ) + r [ X F c , Λ V Λ ′ ] , i.e., D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE C ( X α ) ] = D [ BLUE ℛ ( X F c α ) ] holds. Furthermore, D [ BLUE ℳ ( X α ) ] ≼ D [ BLUE C ( X α ) ] is not provided since i + ( H 1 ) ≠ r [ X , Λ V Λ ′ ] + r ( X F c ) . We also mention that since it is easily seen from the calculations that the difference D [ BLUE ℳ ( X α ) ] − D [ BLUE C ( X α ) ] or D [ BLUE ℳ ( X α ) ] − D [ BLUE ℛ ( X F c α ) ] is psd, namely, we observe that the inequality D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE C ( X α ) ] or D [ BLUE ℳ ( X α ) ] ≽ D [ BLUE ℛ ( X F c α ) ] already holds.

6 Concluding remarks

This article conducts comparisons related to the dispersion matrices of the BLUPs in the context of a CLMM for a general vector of all unknown vectors. The goal is to assess the optimality of these predictors and make comparisons with any symmetric matrix. The methodology employed for these comparisons includes the use of block matrix inertias and ranks, along with EBMO s. By applying these techniques, this article derives various equalities and inequalities that help in evaluating and comparing the performance of predictors within the CLMM . Furthermore, the comparison results obtained for the CLMM can be extended to apply to special instances of the general vector of all unknown vectors, providing a more comprehensive understanding of how these predictors perform in different scenarios. Additionally, this article explores the relationship between the CLMM and its unconstrained counterpart, shedding light on the impact of constraints on prediction and estimation within these models.

In this study, the approach of reparameterization of LMM s subject to exact linear restrictions is used for inference results. Another popular approach to handling a CLMM is to construct a new combined model by merging two given parts of the CLMM , i.e., the model part and the constraint part, into a combined form. According to this approach, the explicitly constrained model C is converted into the following implicitly CLMM :

(42) C ^ : y c = X C α + Z 0 γ + ε 0 , with D y c = σ 2 Λ V Λ ′ 0 0 0 .

Then, by taking into consideration the model C ^ in (42), similar conclusions can be derived for the BLUP of ψ .

Overall, the article’s findings contribute to the field of statistical modeling and analysis by offering insights into the optimality and performance of predictors in the context of CLMM s and their comparisons with other matrices. The additional comparison findings that reduce to special instances of the general vector of all unknown vectors indicate that the insights gained from the CLMM can be applied to specific cases or scenarios within this broader model. This reduction allows for a more targeted understanding of how the CLMM and its predictors perform under particular conditions, potentially providing practical guidance for model selection and estimation. Furthermore, this article provides comparisons between the CLMM and its unconstrained form. This comparison is valuable for understanding the impact of constraints on the model’s performance. It can reveal how imposing certain restrictions on the model’s parameters, as is done in the CLMM , affects the quality of predictions and estimations compared to the unconstrained version of the model. In summary, these comparisons not only offer insights into the CLMM but also allow for the application of these insights in specific scenarios and provide a basis for evaluating the trade-offs between constrained and unconstrained modeling approaches.

Acknowledgement

The authors are grateful to anonymous referees for their careful reading, valuable suggestions, helpful comments on the original version of this article.

Funding information: This work was funded by Sakarya University.
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
Conflict of interest: The authors state no conflicts of interest.

References

[1] H. Haupt and W. Oberhofer, Stochastic response restrictions, J. Multivariate Anal. 95 (2005), no. 1, 66–75, DOI: https://doi.org/10.1016/j.jmva.2004.08.006. 10.1016/j.jmva.2004.08.006Search in Google Scholar

[2] X. Ren, Corrigendum to “On the equivalence of the BLUEs under a general linear model and its restricted and stochastically restricted models” [Statist. Probab. Lett. 90 (2014) 1–10], Statist. Probab. Lett. 104 (2015), 181–185, DOI: https://doi.org/10.1016/j.spl.2015.05.004. 10.1016/j.spl.2015.05.004Search in Google Scholar

[3] A. R. Gallant and T. M. Gerig, Computations for constrained linear models, J. Econometrics 12 (1980), no. 1, 59–84, DOI: https://doi.org/10.1016/0304-4076(80)90053-6. 10.1016/0304-4076(80)90053-6Search in Google Scholar

[4] C. R. Hallum, T. O. Lewis, and T. L. Boullion, Estimation in the restricted general linear model with a positive semidefinite covariance matrix, Comm. Statist. 1 (1973), no. 2, 157–166, DOI: https://doi.org/10.1080/03610927308827014. 10.1080/03610927308827014Search in Google Scholar

[5] N. Güler and M. E. Büyükkaya, Statistical inference of a stochastically restricted linear mixed model, AIMS Math. 8 (2023), no. 10, 24401–24417, DOI: https://doi.org/10.3934/math.20231244. 10.3934/math.20231244Search in Google Scholar

[6] C. A. McGilchrist and C. W. Aisbett, Restricted BLUP for mixed linear models, Biom. J. 33 (1991), no. 2, 131–141, DOI: https://doi.org/10.1002/bimj.4710330202. 10.1002/bimj.4710330202Search in Google Scholar

[7] M. Satoh, An alternative derivation method of mixed model equations from best linear unbiased prediction (BLUP) and restricted BLUP of breeding values not using maximum likelihood, Anim. Sci. J. 89 (2018), no. 6, 876–879, DOI: https://doi.org/10.1111/asj.13016. 10.1111/asj.13016Search in Google Scholar PubMed PubMed Central

[8] B. Jiang and Y. Tian, On best linear unbiased estimation and prediction under a constrained linear random-effects model, J. Ind. Manag. Optim. 19 (2023), no. 2, 852–867, DOI: https://doi.org/10.3934/jimo.2021209. 10.3934/jimo.2021209Search in Google Scholar

[9] Y. Sun, B. Jiang, and H. Jiang, Computations of predictors/estimators under a linear random-effects model with parameter restrictions, Comm. Statist. Theory Methods 48 (2019), no. 14, 3482–3497, DOI: https://doi.org/10.1080/03610926.2018.1476714. 10.1080/03610926.2018.1476714Search in Google Scholar

[10] J. K. Baksalary and R. Kala, Best linear unbiased estimation in the restricted general linear model, Series Statistics 10 (1979), no. 1, 27–35, DOI: https://doi.org/10.1080/02331887908801464. 10.1080/02331887908801464Search in Google Scholar

[11] J. S. Chipman and M. M. Rao, The treatment of linear restrictions in regression analysis, Econometrica 32 (1964), no. 1/2, 198–209, DOI: https://doi.org/10.2307/1913745. 10.2307/1913745Search in Google Scholar

[12] H. Jiang, J. Qian, and Y. Sun, Best linear unbiased predictors and estimators under a pair of constrained seemingly unrelated regression models, Statist. Probab. Lett. 158 (2020), 108669, DOI: https://doi.org/10.1016/j.spl.2019.108669. 10.1016/j.spl.2019.108669Search in Google Scholar

[13] W. Li, Y. Tian, and R. Yuan, Statistical analysis of a linear regression model with restrictions and superfluous variables, J. Ind. Manag. Optim. 19 (2023), no. 5, 3107–3127, DOI: https://doi.org/10.3934/jimo.2022079. 10.3934/jimo.2022079Search in Google Scholar

[14] T. Mathew, A note on best linear unbiased estimation in the restricted general linear model, Series Statistics 14 (1983), no. 1, 3–6, DOI: https://doi.org/10.1080/02331888308801679. 10.1080/02331888308801679Search in Google Scholar

[15] N. Güler and M. E. Büyükkaya, Some remarks on comparison of predictors in seemingly unrelated linear mixed models, Appl. Math. 67 (2022), 525–542, DOI: https://doi.org/10.21136/AM.2021.0366-20. 10.21136/AM.2021.0366-20Search in Google Scholar

[16] Y. Tian, Some equalities and inequalities for covariance matrices of estimators under linear model, Statist. Papers 58 (2017), 467–484, DOI: https://doi.org/10.1007/s00362-015-0707-x. 10.1007/s00362-015-0707-xSearch in Google Scholar

[17] Y. Tian and W. Guo, On comparison of dispersion matrices of estimators under a constrained linear model, Stat. Methods Appl. 25 (2016), 623–649, DOI: https://doi.org/10.1007/s10260-016-0350-2. 10.1007/s10260-016-0350-2Search in Google Scholar

[18] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl. 433 (2010), no. 1, 263–296, DOI: https://doi.org/10.1016/j.laa.2010.02.018. 10.1016/j.laa.2010.02.018Search in Google Scholar

[19] C. R. Rao, Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix, J. Multivariate Anal. 3 (1973), no. 3, 276–292, DOI: https://doi.org/10.1016/0047-259X(73)90042-0. 10.1016/0047-259X(73)90042-0Search in Google Scholar

[20] I. S. Alalouf and G. P. H. Styan, Characterizations of estimability in the general linear model, Ann. Statist. 7 (1979), no. 1, 194–200, DOI: https://www.jstor.org/stable/2958842. 10.1214/aos/1176344564Search in Google Scholar

[21] A. S. Goldberger, Best linear unbiased prediction in the generalized linear regression model, J. Amer. Statist. Assoc. 57 (1962), no. 298, 369–375, DOI: https://doi.org/10.2307/2281645. 10.1080/01621459.1962.10480665Search in Google Scholar

[22] S. Puntanen, G. P. H. Styan, and J. Isotalo, Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty, 1st edn., Springer Berlin, Heidelberg, 2011. 10.1007/978-3-642-10473-2_1Search in Google Scholar

[23] Y. Tian and J. Wang, Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model, Comm. Statist. Theory Methods 49 (2020), no. 5, 1201–1216, DOI: https://doi.org/10.1080/03610926.2018.1554138. 10.1080/03610926.2018.1554138Search in Google Scholar

[24] N. Güler and M. E. Büyükkaya, Inertia and rank approach in transformed linear mixed models for comparison of BLUPs, Comm. Statist. Theory Methods 52 (2023), no. 9, 3108–3123, DOI: https://doi.org/10.1080/03610926.2021.1967397. 10.1080/03610926.2021.1967397Search in Google Scholar

[25] G. K. Robinson, That BLUP is a good thing: The estimation of random effects, Statist. Sci. 6 (1991), no. 1, 15–32, DOI: https://www.jstor.org/stable/2245695. 10.1214/ss/1177011926Search in Google Scholar

[26] J. Jiang, Linear and Generalized Linear Mixed Models and Their Applications: Springer Series in Statistics, 1st edn., Springer, New York, NY, 2007. 10.1007/978-1-0716-1282-8_1Search in Google Scholar

[27] H. Yang, H. Ye, and K. Xue, A further study of predictions in linear mixed models, Comm. Statist. Theory Methods 43 (2014), no. 20, 4241–4252, DOI: https://doi.org/10.1080/03610926.2012.725497. 10.1080/03610926.2012.725497Search in Google Scholar

Received: 2023-12-21

Revised: 2024-11-09

Accepted: 2024-12-03

Published Online: 2024-12-26

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

Articles in the same Issue

https://doi.org/10.1515/math-2024-0114

Keywords for this article

best linear unbiased predictor; comparison; constrained linear mixed model; inertia; rank

Creative Commons

BY 4.0