Home Matrix rank and inertia formulas in the analysis of general linear models
Article Open Access

Matrix rank and inertia formulas in the analysis of general linear models

  • Yongge Tian EMAIL logo
Published/Copyright: February 27, 2017
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

Matrix mathematics provides a powerful tool set for addressing statistical problems, in particular, the theory of matrix ranks and inertias has been developed as effective methodology of simplifying various complicated matrix expressions, and establishing equalities and inequalities occurred in statistical analysis. This paper describes how to establish exact formulas for calculating ranks and inertias of covariances of predictors and estimators of parameter spaces in general linear models (GLMs), and how to use the formulas in statistical analysis of GLMs. We first derive analytical expressions of best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) of all unknown parameters in the model by solving a constrained quadratic matrix-valued function optimization problem, and present some well-known results on ordinary least-squares predictors/ordinary least-squares estimators (OLSPs/OLSEs). We then establish some fundamental rank and inertia formulas for covariance matrices related to BLUPs/BLUEs and OLSPs/OLSEs, and use the formulas to characterize a variety of equalities and inequalities for covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. As applications, we use these equalities and inequalities in the comparison of the covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. The work on the formulations of BLUPs/BLUEs and OLSPs/OLSEs, and their covariance matrices under GLMs provides direct access, as a standard example, to a very simple algebraic treatment of predictors and estimators in linear regression analysis, which leads a deep insight into the linear nature of GLMs and gives an efficient way of summarizing the results.

Keywords: General linear model; Predictor; Estimator; Covariance matrix; Rank; Inertia; Equality; Inequality
MSC 2010: 15A03; 15A09; 62H12; 62J05

1 Introduction

Throughout this paper, the symbol ℝm×n stands for the collection of all m×n real matrices. The symbols 𝐀′, r(𝐀), and ℛ(𝐀) stand for the transpose, the rank, and the range (column space) of a matrix 𝐀 ∈ ℝm×n, respectively; 𝐈m denotes the identity matrix of order m. The Moore–Penrose generalized inverse of 𝐀, denoted by 𝐀+, is defined to be the unique solution 𝐗 satisfying the four matrix equations

AGA=A,GAG=G,(AG)′=AG,(GA)′=GA.

Further, let 𝐏𝐀, 𝐄𝐀 and 𝐅𝐀 stand for the three orthogonal projectors (symmetric idempotent matrices) 𝐏𝐀 = 𝐀𝐀+, 𝐄𝐀 = 𝐀⊥ = 𝐈m - 𝐀𝐀+, and 𝐅𝐀 = 𝐈n - 𝐀+𝐀. All about the orthogonal projectors 𝐏𝐀, 𝐄𝐀, and 𝐅𝐀 with their applications in the linear statistical models can be found in [1–3]. The symbols i+(𝐀) and i−(𝐀) for 𝐀 = 𝐀′ ∈ ℝm×m, called the positive inertia and negative inertia of 𝐀, denote the number of the positive and negative eigenvalues of 𝐀 counted with multiplicities, respectively. For brief, we use i±(𝐀) to denote the both numbers. 𝐀 ≻ 0, 𝐀 ≽ 0, 𝐀 ≺ 0, and 𝐀 ≼ 0 mean that 𝐀 is a symmetric positive definite, positive semi-definite, negative definite, negative semi-definite matrix, respectively. Two symmetric matrices 𝐀 and 𝐁 of the same size are said to satisfy the inequalities 𝐀≻ 𝐁, 𝐀 ≽ 𝐁, 𝐀≺ 𝐁, and 𝐀 ≼ 𝐁 in the Löwner partial ordering if 𝐀 – 𝐁 is positive definite, positive semi-definite, negative definite, and negative semi-definite, respectively. Also, it is well known that the Löwner partial ordering is a surprisingly strong and useful property between two symmetric matrices. For more results on connections between inertias and the Löwner partial ordering of real symmetric (complex Hermitian) matrices, as well as applications of inertias and the Löwner partial ordering in statistics, see, e.g., [2, 4–9].

Recall that linear models are the first type of regression models to be studied extensively in statistical inference, which have had a profound impact in the field of statistics and applications and have been regarded without doubt as a kernel part in current statistical theory. A typical form of linear models is defined by

y=Xβ+ε,E(ε)=0,D(ε)=σ2Σ, (1)

where y ∈ ℝn × 1 is vector of observable response variables, 𝐗 ∈ ℝn × p is a known matrix of arbitrary rank, β ∈ ℝp × 1 is a vector of fixed but unknown parameters, E(𝜺) and D(𝜺) denote the expectation vector and the dispersion matrix of the random error vector 𝜺 ∈ ℝn× 1, 𝚺 ∈ ℝn× n is a known positive semi-definite matrix of arbitrary rank, and σ2 is unknown positive number.

Once a general linear model (GLM) is formulated, the first and most important task is to estimate or predict unknown parameters in the model by using various mathematical and statistical tools. As a foundation of current regression theory, there are a sufficient bunch of results on estimators and predictors of parameter spaces in GLMs. Even so, people are still able to obtain more and more new and valuable results on statistical inference of GLMs. Estimation of β as well as prediction of 𝜺 in (1) are major concerns in the statistical inference of (1), and it is always desirable, as claimed in [10, 11], to simultaneously identify estimators and predictors of all unknown parameters in GLMs. As formulated in [10, 11], a general vector of linear parametric functions involving the two unknown parameter vectors β and 𝜺 in (1) is given by

ϕ=Kβ+Jε, (2)

where 𝐊 ∈ ℝk × p and 𝐉 ∈ ℝk × n are given matrices of arbitrary ranks. Eq. (2) includes all vector and matrix operations in (1) as its special cases. For instance,

  1. if 𝐊 = 𝐗 and 𝐉 = 𝐈n, (2) becomes ϕ = 𝐗 β + 𝜺 = y, the observed response vector;

  2. if 𝐉 = 0, (2) becomes ϕ = 𝐊 β, a general vector of linear parametric functions;

  3. if 𝐊 = 𝐗 and 𝐉 = 0, (2) becomes ϕ = 𝐗 β, the mean vector;

  4. if 𝐊 = 0 and 𝐉 = 𝐈n, (2) becomes ϕ = 𝜺, the random error vector.

Theoretical and applied researches seek to develop various possible predictors/estimators of (2) and its specific cases. During these approaches, the unbiasedness of ϕ with respect to the parameter spaces in (1) is an important property. In statistical inference of a GLM, usually there are many unbiased predictors/estimators for the same parameter space. Under the situation that there exist unbiased predictors/estimators for the same parameter space, it is natural to find such an unbiased predictor/estimator that has the smallest dispersion matrix among all the unbiased predictors/estimators. Thus, the unbiasedness and the smallest dispersion matrices of predictors/estimators are most intrinsic requirements in the statistical analysis of GLMs. Based on these requirements, we introduce the following classic concepts of the predictability, estimability, and the BLUP/BLUEs of ϕ in (2) and its special cases originated from [12].

Definition 1.1

The vector ϕ in (2) is said to be predictable under (1) if there exists an 𝐋 ∈ ℝk × n such that 𝐄(𝐋y – ϕ) = 0. In particular, 𝐊β is said to be estimable under (1) if there exists an 𝐋∈ ℝk × n such that 𝐄(𝐋y – 𝐊β) = 0.

Definition 1.2

Let ϕ be as given in (2) and assume that it is predictable under (1). If there exists a matrix 𝐋 such that

E(Ly−ϕ)=0andD(Ly−ϕ)=min (3)

holds in the Löwner partial ordering, the linear statistic Ly is defined to be the best linear unbiased predictor (BLUP) of ϕ under (1), and is denoted by

Ly=BLUP(ϕ)=BLUP(Kβ+Jε). (4)

If 𝐉 = 0 in (2), or 𝐊 = 0 in (2), then the Ly satisfying (3) is called the best linear unbiased estimator (BLUE) and the BLUP of 𝐊β and 𝐉𝜺 under (1), respectively, and are denoted by

Ly=BLUE(Kβ),Ly=BLUP(Jε), (5)

respectively

Definition 1.3

Let ϕ be as given in (2). The ordinary least-squares estimator (OLSE) of the unknown parameter vector β in (1) is defined to be

OLSE(β)=argminβ∈Rp×1(y−Xβ)′(y−Xβ), (6)

while the OLSE of 𝐊β under (1) is defined to be OLSE(𝐊β) = 𝐊OLSE(β); the ordinary least-squares predictor (OLSP) of the random error vector 𝜺 in (1) is defined to be

OLSP(ε)=y−OLSE(Xβ), (7)

while the OLSP of 𝐉𝜺 under (1) is defined to be OLSP (𝐉𝜺) = 𝐉OLSP(𝜺). The OLSP of ϕ under (1) is defined to be

OLSP(ϕ)=OLSE(Kβ)+OLSP(Jε). (8)

The above definitions enable us to deal with various prediction and estimation problems under most general assumptions of GLMs.

The purpose of this paper is to investigate the performances of BLUP(ϕ), BLUP(ϕ)–ϕ, OLSP(ϕ), and OLSP(ϕ)–ϕ under the assumption that ϕ is predictable under (1) by using the matrix rank/inertia methodology. Our work in this paper includes:

  1. establishing linear matrix equations and exact analytical expressions of the BLUPs and the OLSPs of (2);

  2. characterizing algebraic and statistical properties of the BLUPs and the OLSPs of (2);

  3. deriving formulas for calculating

    i±(A−D[ϕ−BLUP(ϕ)]),r(A−D[ϕ−BLUP(ϕ)]), (9)
    i±(A−D[ϕ−OLSP(ϕ)]),r(A−D[ϕ−OLSP(ϕ)]), (10)

    where A is a symmetric matrix, such as, dispersion matrices of other predictors and estimators;

  4. establishing necessary and sufficient conditions for the following matrix equalities and inequalities

    D[ϕ−BLUP(ϕ)]=A,D[BLUP(ϕ)−ϕ]≻A(≽A,≺A,≼A), (11)
    D[ϕ−OLSP(ϕ)]=A,D[OLSP(ϕ)−ϕ]≻A(≽A,≺A,≼A) (12)

    to hold, respectively, in the Löwner partial ordering;

  5. establishing equalities and inequalities between the BLUPs and OLSPs of ϕ and their dispersion matrices.

As defined in (1), we do not attach any restrictions to the ranks of the given matrices in the model in order to obtain general results on this group of problems. Regression analysis is a very important statistical method that investigates the relationship between a response variable and a set of other variables named as independent variables. Linear regression models were the first type of models to be studied rigorously in regression analysis, which were regarded without doubt as a noble and magnificent part in current statistical theory. As demonstrated in most statistical textbooks, some common estimators of unknown parameters under a linear regression model, such as, the well-known OLSEs and BLUEs are usually formulated from certain algebraic operations of the observed response vector, the given model matrix, and the covariance matrix of the error term in the model. Hence, the standard inference theory of linear regression models can be established without tedious and ambiguous assumptions from the exact algebraic expressions of estimators, which is easily acceptable from both mathematical and statistical points of view. In fact, linear regression models are only type of statistical models that have complete and solid supports from linear algebra and matrix theory, while almost all results on linear regression models can be formulated in matrix forms and calculations. It is just based this fact that linear regression models attract a few of linear algebraists to consider their matrix contributions in statistical analysis.

The paper is organized as follows. In Section 2, we introduce a variety of mathematical tools that can be used to simplify matrix expressions and to characterize matrix equalities, and also present a known result on analytical solutions of a quadratic matrix-valued function optimization problems. In Section 3, we present a group of known results on the predictability of (2), the exact expression of the BLUP of (2) and its special cases, as well as various statistical properties and features of the BLUP. In Section 4, we establish a group of formulas for calculating the rank and inertia in (9) and use the formulas to characterize the equality and inequalities in (11). In Section 5, we first give a group of results on the OLSP of (2) and its statistical properties. We then establish a group of formulas for calculating the rank and inertia in (10) and use the formulas to characterize the equality and inequalities in (12). The connections between the OLSP and BLUP of (2), as well as the equality and inequalities between the dispersion matrices of the OLSP and BLUP of (2) are investigated in Section 6. Conclusions and discussions on algebraic tools in matrix theory, as well as the applications of the rank/inertia formulas in statistical analysis are presented in Section 7.

2 Preliminaries in linear algebra

This section begins with introducing various formulas for ranks and inertias of matrices and claims their usefulness in matrix analysis and statistical theory. Recall that the rank of a matrix and the inertia of a real symmetric matrix are two basic concepts in matrix theory, which are the most significant finite nonnegative integers in reflecting intrinsic properties of matrices, and thus are cornerstones of matrix mathematics. The mathematical prerequisites for understanding ranks and inertias of matrices are minimal and do not go beyond elementary linear algebra, while many simple and classic formulas for calculating ranks and inertias of matrices can be found in most textbooks of linear algebra. The intriguing connections between generalized inverses and ranks of matrices were recognized in 1970s. A variety of fundamental formulas for calculating the ranks of matrices and their generalized inverses were established, and many applications of the rank formulas in matrix theory and statistics were presented in [13]. Since then, the matrix rank theory has been greatly developed and has become an influential and effective tool in simplifying complicated matrix expressions and establishing various matrix equalities.

In order to establish and characterize various possible equalities for predictors and estimators under GLMs, and to simplify various matrix equalities composed by the Moore–Penrose inverses of matrices, we need the following well-known results on ranks and inertias of matrices to make the paper self-contained.

Lemma 2.1

Let 𝐀, 𝐁 ∈ ℝm×n, or 𝐀 = 𝐀′, 𝐁 = 𝐁′ ∈ ℝm × m. Then the following results hold:

  1. 𝐀 = 𝐁 if and only if r(𝐀–𝐁) = 0,

  2. A – B is nonsingular if and only if r(𝐀–𝐁) = m = n,

  3. 𝐀 ≻ 𝐁 (𝐀 ≺ 𝐁) if and only if i+ (𝐀 – 𝐁) = m (i–(𝐀 – 𝐁) = m),

  4. 𝐀 ≽ 𝐁 (𝐀 ≼ 𝐁) if and only if i− (𝐀−𝐁) = 0 (i+(𝐀−𝐁) = 0).

The assertions in Lemma 2.1 directly follow from the definitions of ranks, inertias, definiteness, and semi-definiteness of (symmetric) matrices, and were first summarized and effectively utilized in [4]. This lemma manifests that if certain explicit formulas for calculating ranks/inertias of differences of (symmetric) matrices are established, we can use the formulas to characterize the corresponding matrix equalities and inequalities. Thus there are important and peculiar consequences of establishing formulas for calculating ranks/inertias of matrices. This fact reflects without doubt the most exciting roles of matrix ranks/inertias in matrix analysis and applications, and thus it is technically necessary to establish exact matrix rank/inertia formulas as many as possible from the theoretical and applied points of view.

Lemma 2.2

([13]). Let 𝐀 ∈ ℝm×n, 𝐁 ∈ ℝm × k, and 𝐂 ∈ ℝl × n. Then

r[A,B]=r(A)+r(EAB)=r(B)+r(EBA), (13)
rAC=r(A)+r(CFA)=r(C)+r(AFC), (14)
rABC0=r(B)+r(C)+r(EBAFC). (15)

In particular, the following results hold.

  1. r[A,B]=r(A)⇔R(B)⊆R(A)⇔AA+B=B⇔EAB=0.

  2. rAC=r(A)⇔R(C′)⊆R(A′)⇔CA+A=C⇔CFA=0.

  3. r[A,B]=r(A)+r(B)⇔R(A)∩R(B)={0}⇔R[(EAB)′]=R(B′)⇔R[(EBA)′]=R(A′).

  4. rAC=r(A)+r(C)⇔R(A′)∩R(C′)={0}⇔R(CFA)=R(C)⇔R(AFC)=R(A).

  5. rABC0=r(B)+r(C)⇔EBAFC=0.

  6. r(A+B)=r(A)+r(B)⇔R(A)∩R(B)={0}andR(A′)∩R(B′)={0}.

Lemma 2.3

([14]). If ℛ(𝐀1′) ⊆ ℛ(𝐁1′), ℛ(𝐀2) ⊆ ℛ(𝐁1), ℛ(𝐀2′) ⊆ ℛ(𝐁2′), and ℛ(𝐀3) ⊆ ℛ(𝐁2), then

r(A1B1+A2)=rB1A2A10−r(B1), (16)
r ( A 1 B 1 + A 2 B 2 + A 3 ) = r 0 B 2 A 3 B 1 A 2 0 A 1 0 0 − r ( B 1 ) − r ( B 2 ) . (17)

The results collected in the following lemma are obvious or well known.

Lemma 2.4

Let 𝐀 = 𝐀′ ∈ ℝm × m, 𝐁 = 𝐁′ ∈ ℝn× n, Q ∈ ℝm×n, and assume that 𝐏∈ ℝm × m is nonsingular. Then

r(A)=i+(A)+i−(A), (18)
i±(PAP′)=i±(A)(Sylvester′slawofinertia), (19)
i±(A+)=i±(A),i±(−A)=i∓(A), (20)
i±A00B=i±(A)+i±(B), (21)
i+0QQ′0=i−0QQ′0=r(Q). (22)

Lemma 2.5

([4]). Let 𝐀 = 𝐀′ ∈ ℝm × m, 𝐁∈ ℝm×n, and D = D′ ∈ ℝn× n. Then

i±ABB′0=r(B)+i±(EBAEB),rABB′0=2r(B)+r(EBAEB), (23)

i ± A B B ′ D = i ± ( A ) + i ± 0 E A B B ′ E A D − B ′ , A + B . (24)

In particular,

i+ABB′0=r[A,B],i−ABB′0=r(B),rABB′0=r[A,B]+r(B),ifA≽0, (25)

i±ABB′D=i±(A)+i±(D−B′A+B)ifR(B)⊆R(A). (26)

Lemma 2.6

([4]). Let A = A′ ∊ ℝm × m, B ∊ ℝq × n, D = D′ ∊ ℝn×n, P ∊ ℝq×m with ℛ(A) ⊆ ℛ(P′) and ℛ(B) ⊆ ℛ(P). Also let M=−AP′0P0B0B′D. Then

i±[D−B′(P′)+AP+B]=i±(M)−r(P), (27)

r[D−B′(P′)+AP+B]=r(M)−2r(P). (28)

Hence,

B′(P′)+AP+B=D⇔r(M)=2r(P),B′(P′)+AP+B≽D⇔i+(M)=r(P),B′(P′)+AP+B≼D⇔i−(M)=r(P).

Lemma 2.7

([15]). The linear matrix equation AX = B is consistent if and only if r[A, B] = r(A), or equivalently, AA+B = B. In this case, the general solution of the equation can be written in the parametric form X = A+B+ FAU, where U is an arbitrary matrix.

In statistical inference of parametric regression models, the unknown parameters in them are usually predicted/estimated by various optimization methods or algorithms. A brief survey on modern optimization methods in statistical analysis can be found in [16]. In any case, we expect that the optimization problems occurred in parameter predictions/estimations under a GLM have analytical solutions, so that we can use the analytical solutions to establish a perfect theory on the statistical inference of the GLM. The notion of BLUP was well established in the literature, but it is not easy to believe that there exist analytical results on BLUPs of a very general nature due to the lack of explicit solutions of BLUPs’ optimization problems. The present author recently developed an algebraic method of solving quadratic matrix-valued function optimization problems in [17], and used the method to derive many new analytical matrix equations and formulas of BLUPs of all unknown parameters in GLMs with random effects. By using this optimization method, we are now able to directly obtain exact formulas for calculating the BLUPs of ϕ in (2) and their dispersion matrices, and to use the formulas in dealing with various statistical inference problems under general assumptions. In order to directly solve the matrix minimization problem associated with the BLUPs under a GLM, we need the following known result on analytical solutions of a constrained quadratic matrix-valued function minimization problem.

Lemma 2.8

([17]). Let

f(L)=(L+C)M(L+C)′s.t.LA=B,

where A ∊ ℝp×q, B ∊ ℝn×q, and C ∊ ℝn×p are given, M ∊ ℝp×p is positive semi-definite, and the matrix equation LA = B is consistent. Then there always exists a solution L0 of L0A = B such that

f(L)≽f(L0)

holds for all solutions of LA = B, while the matrix L0 satisfying the inequality is determined by the following consistent matrix equation

L0[A,MA⊥]=[B,−CMA⊥].

In this case, the general expression of L0 and the corresponding f(L0) and f(L)–f(L0) are given by

L0=argminLA=B⁡f(L)=[B,−CMA⊥][A,MA⊥]++U[A,M]⊥,

f(L0)=minLA=Bf(L)=KMK′−KM(A⊥MA⊥)+MK′,f(L)−f(L0)=(LMA⊥+CMA⊥)(A⊥MA⊥)+(LMA⊥+CMA⊥)′,

where K = BA+ + C and U ∊ ℝn× p is arbitrary.

The assertions in this lemma show that a clear way of obtaining analytical solutions of a typical constrained quadratic matrix-valued function minimization problem is established, and all the properties and features of the minimization problem can been discovered from the analytical solutions. With the supports of the assertions in Lemma 2.1 and the rank/inertia formulas in Lemmas 2.2–2.6, we are able to convert many inference problems in statistics into algebraic problems of characterizing matrix equalities and inequalities composed by matrices and their generalized inverses, and to derive, as demonstrated in Sections 4–6 below, analytical solutions of the problems by using the methods of matrix equations, matrix rank formulas, and various tricky partitioned matrix calculations.

3 Formulas and properties of BLUPs

In what follows, we assume that (1) is consistent, that is, y ∊ ℛ[X, Σ] holds with probability 1; see [18, 19].

Notice from (3) that the BLUP of ϕ in (2) is defined from the minimization of the dispersion matrix of Ly–ϕ subject to E(Ly–ϕ) = 0. The advantage of the minimization property of the BLUP has created many problems in statistical inference of GLMs. In particular, the minimum dispersion matrix of Ly–ϕ can be utilized to compare the optimality and efficiency of other types of predictor of ϕ under (1). In fact, BLUPs are primary choices in all possible predictors due to their simple and optimality properties, and have wide applications in both pure and applied disciplines of statistical inference. The theory of BLUPs under GLMs belongs to the classical methods of mathematical statistics, and has been core issues of researches in the field of statistics and applications. It should be pointed out that (3) can equivalently be assembled as certain constrained matrix-valued function optimization problem in the Löwner partial ordering. This kind of equivalences between dispersion matrix minimization problems and matrix-valued function minimization problems were firstly characterized in [11]; see also, [20, 21]. Along with some new development of optimization methods in matrix theory, it is now easy to deal with various complicated matrix operations occurring in the statistical inference of (2).

We next show how to translate the above statistical problems under GLMs into mathematical problems on matrix analysis, and solve the problems by using various results and methods in matrix algebra. Under (1) and (2),

E(ϕ)=Kβ,D(ϕ)=σ2JΣJ′,Cov{ϕ,y}=σ2JΣ, (29)

while Ly–ϕ can be rewritten as

Ly−ϕ=LXβ+Lε−Kβ−Jε=(LX−K)β+(L−J)ε. (30)

Then, the expectation of Ly–ϕ can be expressed as

E(Ly−ϕ)=E[(LX−K)β]=(LX−K)β; (31)

the matrix mean square error of Ly–ϕ is

E[(Ly−ϕ)(Ly−ϕ)′]=(LX−K)ββ′(LX−K)′+σ2(L−J)Σ(L−J)′; (32)

the dispersion matrix of Ly–ϕ is

D(Ly−ϕ)=σ2(L−J)Σ(L−J)′=△σ2f(L). (33)

Hence, the constrained covariance matrix minimization problem in (3) converts to a mathematical problem of minimizing the quadratic matrix-valued function f(L) subject to (LX–K)β = 0. A general method for solving this kind of matrix optimization problems in the Löwner partial ordering was formulated in Lemma 2.8. In particular, the following comprehensive results were established in [22]; see also [23].

Theorem 3.1

The vector ϕ in (2) is predictable by y in (1) if and only if

R(X′)⊇R(K′). (34)

Proof

It follows from (31) that E (Ly - ϕ) = 0⇔(LX - K)β = 0 for all β ⇔ LX = K. From Lemma 2.7, the matrix equation is consistent if and only if (34) holds.

Theorem 3.2

(Fundamental BLUP equation). Assume that ϕ in (2) is predictable under (1). Then

E(Ly−ϕ)=0andD(Ly−ϕ)=min⇔L[X,ΣX⊥]=[K,JΣX⊥]. (35)

The equation in (35), called the fundamental BLUP equation, is consistent, i.e.,

[K,JΣX⊥][X,ΣX⊥]+[X,ΣX⊥]=[K,JΣX⊥] (36)

holds under (34), while the general expression of L and the corresponding BLUP (ϕ) are given by

BLUP ( ϕ ) = L y = [ K , J Σ X ⊥ ] [ X , Σ X ⊥ ] + + U [ X , Σ X ⊥ ] ⊥ y , (37)

where U ∊ ℝk×n is arbitrary. In particular,

BLUE(Kβ)=([K,0][X,ΣX⊥]++U1[X,ΣX⊥]⊥)y, (38)

BLUP(Jε)=([0,JΣX⊥][X,ΣX⊥]++U2[X,ΣX⊥]⊥)y, (39)

where U1, U2 ∊ ℝk×n are arbitrary. Furthermore, the following results hold.

  1. r[X, Σ X⊥] = r[X, Σ], ℛ[X, Σ X⊥] = ℛ[X, Σ], and ℛ(X)∩ ℛ(Σ X⊥) = {0}.

  2. L is unique if and only if r[X, Σ] = n.

  3. BLUP (ϕ) is unique with probability 1 if and only if (1) is consistent.

  4. The dispersion matrices of BLUP (ϕ) and ϕ-BLUP(ϕ), as well as the covariance matrix between BLUP (ϕ) and ϕ are unique, and satisfy the following equalities

    D[BLUP(ϕ)]=σ2[K,JΣX⊥][X,ΣX⊥]+Σ([K,JΣX⊥][X,ΣX⊥]+)′, (40)

    Cov{BLUP(ϕ),ϕ}=σ2[K,JΣX⊥][X,ΣX⊥]+ΣJ′, (41)

    D(ϕ)−D[BLUP(ϕ)]=σ2JΣJ′−σ2[K,JΣX⊥][X,ΣX⊥]+Σ([K,JΣX⊥][X,ΣX⊥]+)′, (42)

    D[ϕ−BLUP(ϕ)]=σ2(J−[K,JΣX⊥][X,ΣX⊥]+)Σ(J−[K,JΣX⊥][X,ΣX⊥]+)′. (43)

  5. BLUP (ϕ), BLUE (K β ), and BLUP (J𝜺) satisfy

    BLUP(ϕ)=BLUE(Kβ)+BLUP(Jε), (44)

    Cov{BLUE(Kβ),BLUP(Jε)}=0, (45)

    D[BLUP(ϕ)]=D[BLUE(Kβ)]+D[BLUP(Jε)]. (46)

  6. Tϕ is predictable for any matrix T ∊ ℝt× k, and BLUP (Tϕ)=TBLUP(ϕ) holds.

  7. In particular,

    BLUE(Xβ)=([X,0][X,ΣX⊥]++U[X,ΣX⊥]⊥)y, (47)

    D[BLUE(Xβ)]=σ2[X,0][X,ΣX⊥]+Σ([X,0][X,ΣX⊥]+)′, (48)

    where U ∊ ℝn×n is arbitrary, and

    BLUP(ε)=([0,ΣX⊥][X,ΣX⊥]++U[X,ΣX⊥]⊥)y=(Σ(X⊥ΣX⊥)++U[X,ΣX⊥]⊥)y, (49)

    Cov{BLUP(ε),ε}=D[BLUP(ε)]=σ2Σ(X⊥ΣX⊥)+Σ, (50)

    D[ε−BLUP(ε)]=D(ε)−D[BLUP(ε)]=σ2Σ−σ2Σ(X⊥ΣX⊥)+Σ, (51)

    where U ∊ ℝn×n is arbitrary.

  8. y, BLUE (X β ), and BLUP (𝜺) satisfy

    y=BLUE(Xβ)+BLUP(ε), (52)

    Cov{BLUE(Xβ),BLUP(ε)}=0, (53)

    D(y)=D[BLUE(Xβ)]+D[BLUP(ε)]. (54)

Proof

Eq. (3) is equivalent to finding a solution L0 of L0X=K such that

f(L)≽f(L0)s.t.LX=K (55)

holds in the Löwner partial ordering. From Lemma 2.8, there always exists a solution L0 of L0X=K such that f(L)≽ f(L0) holds for all solutions of LX=K, and the L0 is determined by the matrix equation L0[X, Σ X⊥]= [K, JΣ X⊥], establishing the matrix equation in (35). Solving the matrix equation in (35) yields (37). Eqs. (38) and (39) follow directly from (37).

The three formulas in (a) are well known for the coefficient matrix in (37); see e.g., [1] and [2, p. 123]. Note that [X, Σ X⊥]⊥=0⇔[X, Σ X⊥][X, Σ X⊥]+=In⇔ r[X, Σ X⊥]=r[X, Σ]=n. Combining this fact with (37) leads to (b). Setting the term [X, Σ X⊥]⊥y = 0 in (37) leads to (c). Eqs. (40)–(43) follow from (1), (29), (33), and (37).

Rewrite (37) as

BLUP(ϕ)=([K,JΣX⊥][X,ΣX⊥]++U[X,ΣX⊥]⊥)y=([K,0][X,ΣX⊥]++U1[X,ΣX⊥]⊥)y+([0,JΣX⊥][X,ΣX⊥]++U2[X,ΣX⊥]⊥)y=BLUE(Kβ)+BLUP(Jε),

establishing (44).

From (38) and (39), the covariance matrix between BLUE (K β ) and BLUP (J𝜺) is

Cov{BLUE(Kβ),BLUP(Jε)}=σ2([K,0][X,ΣX⊥]++U1[X,ΣX⊥]⊥)Σ([0,JΣX⊥][X,ΣX⊥]++U2[X,ΣX⊥]⊥)′=σ2[K,0][X,ΣX⊥]+Σ([0,JΣX⊥][X,ΣX⊥]+)′.

Applying (17) to this matrix and simplifying, we obtain

r(Cov{BLUE(Kβ),BLUP(Jε)})=r([K,0][X,ΣX⊥]+Σ([0,JΣX⊥][X,ΣX⊥]+)′)=r 0X′X⊥Σ 0X⊥ΣJ′[X,ΣX⊥] Σ 0 [K,0] 0 0−2r[X,ΣX⊥]=r000−X⊥ΣX⊥X′00X⊥ΣJ′[X,0]Σ0[K,0]00−2r[X,Σ]=r0XKX′Σ0+r[X⊥ΣX⊥,X⊥ΣJ′]−2r[X,Σ]=rXK+rX′Σ+r[X,ΣX⊥,ΣJ′]−r(X)−2r[X,Σ](by(13)and(25))=r(X)+r[X,Σ]+r[X,Σ]−r(X)−2r[X,Σ](by(34))=0,

establishing (45). Eq. (46) follows from (44) and (45). Result (f) follows directly from (37). Results (g) and (h) follow from (38), (39), (d) and (e).

Eq. (35) shows that the BLUPs of all unknown parameters in (2) can jointly be determined by a linear matrix equation composed by the two given coefficient matrices K and J, the given model matrix X, the dispersion matrix of the observed random vector y, and the covariance matrix between ϕ and y. So that we find it convenient to present a simple yet general algebraic treatment of the BLUPs of all unknown parameters in a GLM via a basic linear matrix equation, while the BLUPs have a large number of algebraic and statistical properties and features that are technically convenient from analytical solutions of the matrix equation. Matrix equations and formulas for BLUPs like those in (35) and (37) under GLMs were established in the statistical literature by using various direct and indirect methods, for instance, the BLUE of Kβ and the BLUP of J𝜺, as well as (52) were established separately in [11]. In comparison, the whole results collected in Theorem 3.2 provide an exclusive and unified theory about BLUPs and BLUEs of parameter spaces and their magnificent properties and features under GLMs. As demonstrated in [22, 23], the results in Theorem 3.2 can serve as basic and useful references being applied in the statistical inference of GLMs.

From the fundamental matrix equation and formulas in Theorem 3.2, we are now able to derive many new and valuable consequences on properties of BLUPs of parameter spaces in GLMs under various assumptions. For instance, one of the well-known special cases of (1) is

y=Xβ+ε,E(ε)=0,D(ε)=σ2In, (56)

where σ2 is unknown positive scalar. In this setting, the BLUP and the OLSP of ϕ in (2) coincide under (56), while Theorem 3.2 reduces to the following results.

Corollary 3.3

Assume that ϕ is predictable under (56), i. e., ℛ(X′)⊇ ℛ(K′) holds. Then

E(Ly−ϕ)=0andD(Ly−ϕ)=min⇔L[X,X⊥]=[K,JX⊥].

The matrix equation on the right-hand side is consistent as well, and the unique solution of L and the corresponding Ly are given by

BLUP(ϕ)=Ly=(KX++JX⊥)y.

In particular,

BLUE(Kβ)=KX+y,BLUP(Jε)=JX⊥y,BLUE(Xβ)=XX+y,BLUP(ε)=X⊥y.

Furthermore, the following results hold.

  1. BLUP (ϕ) satisfies the following covariance matrix equalities

    D [ BLUP ( ϕ! ) ] = σ 2 ( K X + + J X ⊥ ) ( K X + + J X ⊥ ) ′ , Cov { BLUP ( ϕ ) , ϕ } = σ 2 ( K X + + J X ⊥ ) J ′ , D ( ϕ ) − D [ BLUP ( ϕ ) ] = σ 2 J J ′ − σ 2 ( K X + + J X ⊥ ) ( K X + + J X ⊥ ) ′ , D [ ϕ − BLUP ( ϕ ) ] = σ 2 ( K X + − J P X ) ( K X + − J P X ) ′ .

  2. BLUP (ϕ), BLUE (Kβ), and BLUP (J𝜺) satisfy

    BLUP(ϕ)=BLUE(Kβ)+BLUP(Jε),Cov{BLUE(Kβ),BLUP(Jε)}=0,D[BLUP(ϕ)]=D[BLUE(Kβ)]+D[BLUP(Jε)].

  3. BLUP (Tϕ) = TBLUP(ϕ) holds for any matrix T ∈ ℝt × k.

  4. BLUE (Xβ) and BLUP (𝜺) satisfy

    BLUE(Xβ)=PXyBLUP(ε)=(In−PX)y,D[BLUE(Xβ)]=σ2PX,Cov{BLUP(ε),ε}=D[BLUP(ε)]=σ2(In−PX),D[ε−BLUP(ε)]=D(ε)−D[BLUP(ε)]=σ2PX.

  5. y, BLUE (Xβ), and BLUP (𝜺) satisfy

    y=BLUE(Xβ)+BLUP(ε),Cov{BLUE(Xβ),BLUP(ε)}=0,D(y)=D[BLUE(Xβ)]+D[BLUP(ε)].

4 Rank/inertia formulas for dispersion matrices of BLUPs

Once predictors/estimators of parameter spaces in GLMs are established, more attention is paid to investigating algebraical and statistical properties and features of the predictors/estimators. Since BLUPs/BLUEs are fundamental statistical methodology to predict and estimate unknown parameters under GLMs, they play an important role in statistical inference theory, and are often taken as cornerstones for comparing the efficiency of different predictors/estimators due to the minimization property of the BLUPs’/BLUEs’ dispersion matrices. As demonstrated in Section 3, we are now able to give exact expressions of BLUPs/BLUEs under GLMs, so that we can derive various algebraical and statistical properties of BLUPs/BLUEs and utilize the properties in the inference of GLMs. Since dispersion matrix of random vector is a conceptual foundation in statistical analysis and inference, statisticians are interested in studying dispersion matrices of predictors/estimators and their algebraic properties. Some previous and present work on the dispersion matrices of BLUPs/BLUEs and their properties under GLMs can be found, e.g., in [24–27]. As is known to all, equalities and inequalities of dispersion matrices of predictors/estimators under GLMs play an essential role in characterizing behaviors of the predictors/estimators. Once certain equalities and inequalities are established for dispersion matrices of predictors/estimators under various assumptions, we can use them to describe performances of the predictors/estimators. This is, however, not an easy task from both mathematical and statistical points of view, because dispersion matrices of predictors/estimators often include various complicated matrix operations of given matrices and their generalized inverses in GLMs, as formulated in (40) –(43). In recent years, the theory of matrix ranks and inertias have been introduced to the statistical analysis of GLMs. We are able to establish various equalities and inequalities for dispersion matrices of predictors/estimators under GLMs by using the matrix rank/inertia methodology; see [5, 6, 9]. Note from (3) that D[ϕ − BLUP(ϕ)] has the smallest dispersion matrix among all Ly – ϕ with E(Ly – ϕ) = 0. So that the dispersion matrix plays a key role in characterizing performances of the BLUP of ϕ in (2). In order to establish possible equalities and inequalities for dispersion matrices of BLUPs, we first establish three basic formulas for calculating the rank/inertia described in (9).

Theorem 4.1

Assume that ϕ in (2) is predictable under (1), and let BLUP (ϕ) be as given in (37). Also let A = A′ ∈ ℝk×k, and denote

M = σ 2 Σ 0 x 0 − A K − J X X ′ K ′ − X ′ J ′ 0 .

Then

i+(A−D[ϕ−BLUP(ϕ)])=i−(M)−r(X), (57)

i−(A−D[ϕ−BLUP(ϕ)])=i+(M)−r[X,Σ], (58)

r(A−D[ϕ−BLUP(ϕ)])=r(M)−r[X,Σ]−r(X). (59)

In consequence, the following results hold.

  1. D[ϕ − BLUP(ϕ)]≻ A ⇔ i+(M) = r(X) + Σ] + k.

  2. D[ϕ − BLUP(ϕ)]≺ A ⇔ i−(M) = r(X) + k.

  3. D[ϕ − BLUP(ϕ)] ≽ A⇔ i−(M)= r(X).

  4. D[ϕ − BLUP(ϕ)]≼ A⇔ i+(M)= r[X, Σ].

  5. D[ϕ − BLUP(ϕ)] = A⇔ r(M) = r[X, Σ] + r(X).

Proof

Note from (43) that

A−D[ϕ−BLUP(ϕ)]=A−σ2([K,JΣX⊥][X,ΣX⊥]+−J)Σ([K,JΣX⊥][X,ΣX⊥]+−J)′=σ2σ−2A−[K,JΣX⊥][X,ΣX⊥]+−J)Σ([K,JΣX⊥][X,ΣX⊥]+−J′. (60)

Also note that ℛ([K, JΣ X⊥]′) ⊆ ℛ([X, Σ X⊥]′) and ℛ(Σ)⊆ ℛ[X, Σ X⊥]. Then applying (26) to (60) and simplifying by Lemmas 2.4 and 2.5, and congruence operations, we obtain

i±(A−D[ϕ−BLUP(ϕ)])=i±σ−2A−[K,JΣX⊥][X,ΣX⊥]+−J)Σ([K,JΣX⊥][X,ΣX⊥]+−J′=i±ΣΣ([K,JΣX⊥][X,ΣX⊥]+−J)′([K,JΣX⊥][X,ΣX⊥]+−J)Σσ−2A−i±(Σ)=i±Σ−ΣJ′−JΣσ−2A+Σ00[K,JΣX⊥]0[X,ΣX⊥][X,ΣX⊥]′0+Σ00[K,JΣX⊥]′−i±(Σ)=i±0−X−ΣX⊥Σ0−X′000K′−X⊥Σ000X⊥ΣJ′Σ00Σ−ΣJ′0KJΣX⊥−JΣσ−2A−i∓0[X,ΣX⊥][X,ΣX⊥]′0−i±(Σ)=i±1−Σ−X−ΣX⊥0ΣJ′−X′000K′−X⊥Σ000X⊥ΣJ′000Σ0JΣKJΣX⊥0σ−2A−JΣJ′−r[X,ΣX⊥]−i±(Σ)(by(22))=i±−Σ−X−ΣX⊥ΣJ′X′00K′−X⊥Σ00X⊥ΣJ′JΣKJΣX⊥σ−2A−JΣJ′−r[X,Σ](by(21))=i±−Σ−X00−X′00K′−X′J′00X⊥ΣX⊥00K−JX0σ−2A−r[X,Σ]=i±−Σ0X0σ−2AJX−KX′X′J′−K′0+i±(X⊥ΣX⊥)−r[X,Σ](by(21))=i∓σ2Σ0X0−AK−JXX′K′−X′J′0+i±(X⊥ΣX⊥)−r[X,Σ](by(20)), (61)

that is,

i+(A−D[ϕ−BLUP(ϕ)])=i−(M)+i+(X⊥ΣX⊥)−r[X,Σ]=i−(M)+r(X⊥Σ)−r[X,Σ]=i−(M)−r(X),i−(A−D[ϕ−BLUP(ϕ)])=i+(M)+i−(X⊥ΣX⊥)−r[X,Σ]=i+(M)−r[X,Σ]

by (13), establishing (57) and (58). Adding the two equalities in (57) and (58) yields (59). Applying Lemma 2.1 to (57)–(59) yields (a)–(e).

Eqs. (57)–(59) establish links between the dispersion matrices of the BLUPs of ϕ and any symmetric matrix. Hence, they can be applied to characterize behaviors of the BLUPs, especially, they can be used to establish many equalities and inequalities for BLUPs’ dispersion matrices under various assumptions. Because the five matrices A, K, J, X, and Σ occur separately in the symmetric block matrix M in Theorem 4.1, it is easy to further simplify (57)–(59) for different choices of the five matrices in the formulas. We next present several special cases of (57)–(59) and give lots of interesting consequences on dispersion matrices of BLUPs/BLUEs and their operations.

Concerning the ranks and inertias of D[ϕ − BLUP(ϕ)] and D(ϕ) − D[BLUP(ϕ)] in (42) and (43), we have the following result.

Theorem 4.2

Assume that ϕ in (2) is predictable under (1), and let

M=−Σ0X000Σ0X0X′000K′0X′00X′J′00KJX0.

Then

r(D[ϕ−BLUP(ϕ)])=rΣXJΣK−r[X,Σ], (62)

and

i±(D(ϕ)−D[BLUP(ϕ)])=i±(M)−r(X)−r[X,Σ], (63)

r(D(ϕ)−D[BLUP(ϕ)])=r(M)−2r(X)−2r[X,Σ]. (64)

Proof

Eq. (62) follows from (59) by setting A = 0. Note from (42) that D(ϕ)−D[BLUP(ϕ)]=σ2JΣJ′−σ2[K,JΣX⊥][X,ΣX⊥]+Σ([K,JΣX⊥][X,ΣX⊥]+)′. Applying (27) and simplifying by congruence transformations gives

i±(D(ϕ)−D[BLUP(ϕ)])=i±(JΣJ′−[K,JΣX⊥][X,ΣX⊥]+Σ([K,JΣX⊥][X,ΣX⊥]+)′)=i±−ΣXΣX⊥0X′00K′X⊥Σ00X⊥ΣJ′0KJΣX⊥JΣJ′−r[X,ΣX⊥]=i±−ΣX00X′00K′00X⊥ΣX⊥X⊥ΣJ′0KJΣX⊥JΣJ′−r[X,Σ]=i±−ΣX000X′00K′000ΣΣJ′X0KJΣJΣJ′000X′00−r(X)−r[X,Σ](by(23))=i±−ΣX000X′00K′000Σ0X0K00−JX00X′−X′J′0−r(X)−r[X,Σ]=i±−Σ0X000Σ0X0X′000K′0X′00−X′J′00K−JX0−r(X)−r[X,Σ]=i±−Σ0X000Σ0X0X′000K′0X′00X′J′00KJX0−r(X)−r[X,Σ]=i±(M)−r(X)−r[X,Σ],

establishing (63) and (64).

Many consequences can be derived from the previous two theorems for different choices of K, J, and A in them. Here, we only give the rank/inertia formulas for the difference D(Ay) − D[ϕ − BLUP(ϕ)].

Corollary 4.3

Assume that ϕ in (2) is predictable under (1), and let BLUP (ϕ) be as given in (37). Also let A ∈ ℝk × n, and denote

M=∑0X0−A∑A′K−JXX′K′−X′J′0.

Then

i + ( D ( A y ) − D [ ϕ − B L U P ( ϕ ) ] ) = i − ( M ) − r ( X ) , i − ( D ( A y ) − D [ ϕ − B L U P ( ϕ ) ] ) = i + ( M ) − r [ X , Σ ] , r ( D ( A y ) − D [ ϕ − B L U P ( ϕ ) ] ) = r ( M ) − r [ X , Σ ] − r ( X ) .

5 Formulas and properties of OLSPs

The method of least squares in statistics is a standard approach for estimating unknown parameters in linear statistical models, which was first proposed as an algebraic procedure for solving overdetermined systems of equations by Gauss (in unpublished work) in 1795 and independently by Legendre in 1805, as remarked in [28–31]. The notion of least-squares estimation is well established in the literature, and we briefly review the derivations of OLSEs and OLSPs. It is easy to verify that the norm (y ‒ Xβ)′(y − X β ) in (6) can be decomposed as the sum

(y−Xβ)′(y−Xβ)=y′EXy+(PXy−Xβ)′(PXy−Xβ),

where y′EXy ⩾ 0 and (PXy − Xβ)′(PXy − Xβ) ⩾ 0. Hence,

minβ∈Rp×1(y−Xβ)′(y−Xβ)=y′EXy+minβ∈Rp×1(PXy−Xβ)′(PXy−Xβ);

see also [7, 8]. The matrix equation Xβ = PXy, which is equivalent to the so-called normal equation X′Xβ = X′y by pre-multiplying X′, is always consistent; see, e.g., [32, p. 114] and [33, pp. 164–165]. Solving this linear matrix equation by Lemma 2.7 yields the following general results.

Theorem 5.1

Assume that ϕ in (2) is predictable by y in (1), and K β is estimable under (1). Then the OLSPs and OLSEs of ϕ and its special cases are given by

OLSE(Kβ)=KX+y, (65)

OLSE(Xβ)=XX+y, (66)

OLSP(ε)=(In−XX+)y=X⊥y, (67)

OLSP(Jε)=JX⊥y, (68)

OLSP(ϕ)=(KX++JX⊥)y. (69)

Furthermore, the following results hold.

  1. OLSP (ϕ) satisfies the following equalities

    D[OLSP(ϕ)]=σ2(KX++JX⊥)Σ(KX++JX⊥)′, (70)

    Cov{OLSP(ϕ),ϕ}=σ2(KX++JX⊥)ΣJ′, (71)

    D(ϕ)−D[OLSP(ϕ)]=σ2JΣJ′−σ2(KX++JX⊥)Σ(KX++JX⊥)′, (72)

    D[ϕ−OLSP(ϕ)]=σ2(KX+−JPX)Σ(KX+−JPX)′. (73)

  2. OLSP (ϕ), OLSE (Kβ), and OLSP (J𝜺) satisfy

    OLSP(ϕ)=OLSE(Kβ)+OLSP(Jε), (74)

    Cov{OLSE(Kβ),OLSP(Jε)}=σ2KX+ΣX⊥J′. (75)

  3. OLSP (Tϕ) = TOLSP(ϕ) holds for any matrix T ∈ ℝt × k.

  4. y, OLSE (Xβ), and OLSP (𝜺) satisfy

    y=OLSE(Xβ)+OLSP(ε), (76)

    Cov{OLSE(Xβ),OLSP(ε)}=σ2PXΣX⊥, (77)

    D[OLSE(Xβ)]=D[ε−OLSP(ε)]=σ2PXΣPX, (78)

    D[OLSP(ε)]=σ2X⊥ΣX⊥, (79)

    D(ε)−D[OLSP(ε)]=σ2Σ−σ2X⊥ΣX⊥. (80)

We next give a group of rank/inertia formulas related to OLSPs and their consequences.

Theorem 5.2

Assume that ϕ in (2) is predictable under (1), and let OLSP (ϕ) be as given in (69). Also let A = A′ ∈ ℝk × k, and denote

M=σ2X′ΣX0X′X0−AK−JXX,XK′−X′J′0.

Then

i±{A−D[ϕ−OLSP(ϕ)]}=i∓(M)−r(X), (81)

r { A − D [ ϕ − O L S P ( ϕ ) ] } = r ( M ) − 2 r ( X ) . (82)

In consequence, the following results hold.

  1. D[ϕ−OLSP(ϕ)]≻A⇔i+(M)=r(X)+k.

  2. D[ϕ−OLSP(ϕ)]≺A⇔i−(M)=r(X)+k.

  3. D[ϕ−OLSP(ϕ)]≽A⇔i−(M)=r(X).

  4. D[ϕ−OLSP(ϕ)]≼A⇔⇔i+(M)=r(X).

  5. D[ϕ−OLSP(ϕ)]=A⇔r(M)=2r(X).

Proof

Note from (73) that

A−D[ϕ−OLSP(ϕ)]=A−σ2(KX+−JPX)Σ(KX+−JPX)′=A−σ2(K−JX)X+Σ(X′)+(K′−X′J′)=A−σ2(K−JX)(X′X)+X′ΣX(X′X)+(K′−X′J′). (83)

Applying (27) to (83) gives

i±{A−D[ϕ−OLSP(ϕ)]}=i±[A−σ2(K−JX)(X′X)+X′ΣX(X′X)+(K′−X′J′)]=i∓σ2X′ΣXX′X0X′X0K′−X′J′0K′−JXA−r(X),=i∓σ2X′ΣX0X′X0−AK−JXX,XK′−X′J′0−r(X),

establishing (81) and (82). Applying Lemma 2.1 to (81) and (82) yields the results in (a)–(e).

Theorem 5.3

Assume that ϕ in (2) is predictable under (1), and let OLSP (ϕ) be as given in (69). Then y, OLSE (Xβ), and OLSP (𝜺)satisfy

r{D(y)−D[OLSE(Xβ)]−D[OLSP(ε)]}=r{D[ε−OLSP(ε)]−D(ε)+D[OLSP(ε)]}=2r[X,ΣX]−2r(X), (84)

r(Cov{OLSE(Xβ),OLSP(ε)})=r[X,ΣX]−r(X). (85)

Proof

It comes from (78)–(80) that

D(y)−D[OLSE(Xβ)]−D[OLSP(ε)]=D(ε)−D[ε−OLSP(ε)]−D[OLSP(ε)]=Cov{OLSE(Xβ),OLSP(ε)}+Cov{OLSP(ε),OLSE(Xβ)}=σ2PXΣX⊥+σ2X⊥ΣPX.

Then by (13) and Lemma 2.2(f),

r{D(y)−D[OLSE(Xβ)]−D[OLSP(ε)]}=r{D[ε−OLSP(ε)]−D(ε)+D[OLSP(ε)]}=r(PXΣX⊥+X⊥ΣPX)=r(PXΣX⊥)+r(X⊥ΣPX)=2r(X⊥ΣPX)=2r[X,ΣPX]−2r(X)=2r[X,ΣX]−2r(X),

as required for (84). By (77) and (13),

r(Cov{OLSE(Xβ),OLSP(ε)})=r(X⊥ΣPX)=r[X,ΣX]−r(X),

as required for (85).

6 Twists of BLUPs and OLSPs under GLMs

After establishing BLUPs and OLSPs of all the unknown parameter vectors in (1) and their algebraic and statistical properties in the previous three sections, it may be of interest to consider the relations between the BLUPs and OLSPs and to establish possible equalities of these predictors. In fact, the equivalence problems of the OLSEs and BLUEs in linear regression theory were initialized and approached in late 1940s from theoretical and applied points of view, which was previously reviewed in [34], and considerable literature exists with many results and discussions on this kind of classic problems; see, e.g., [2, 27, 35–58]. After the sufficient preparations in the previous sections, we are now able to establish possible equalities between BLUPs and OLSPs under various assumptions.

Theorem 6.1

Assume that ϕ in (2) is predictable under (1), and let BLUP (ϕ) and OLSP (ϕ) be as given in (37) and (69), respectively Then the following results hold.

  1. The following statements are equivalent:

    1. OLSP (ϕ) = BLUP(ϕ).

    2. (K − JX)X+ Σ X⊥ = 0.

    3. rX′XX′ΣX⊥K−JX0=r(X).

    4. rX′XX′Σ0X′K−JX0=2r(X).

    5. ℛ{[(K − JX)X+Σ]′} ⊆ ℛ(X).

    6. ℛ([K − JX, 0 ⊆ ℛ([X′X, X′Σ X⊥]′).

    7. R([K−JX,0]′)⊆RX′XX′Σ0X′.

  2. [47, 55] The following statements are equivalent:

    1. OLSE (Kβ) = BLUE(Kβ).

    2. KX+ Σ X⊥ = 0.

    3. rX′XX′ΣX⊥K0=r(X).

    4. rX′XX′Σ0X′K0=2r(X).

    5. ℛ[(KX+Σ)′] ⊆ ℛ(X).

    6. ℛ([K, 0]′) ⊆ ℛ([X′X, X′Σ X⊥]′).

    7. R([K,0]′)⊆RX′XX′Σ0X′.

  3. The following statements are equivalent:

    1. OLSP (J𝜺) = BLUP(J𝜺).

    2. JXX+ Σ X⊥ = 0.

    3. rX′XX′ΣX⊥JX0=r(X).

    4. rX′XX′Σ0X′JX0=2r(X).

    5. ℛ[(JXX+Σ)′] ⊆ ℛ(X).

    6. ℛ([JX, 0]′) ⊆ ℛ([X′X, X′ Σ X⊥]′).

    7. R([JX,0]′)⊆RX′XX′Σ0X′′.

  4. [34] The following statements are equivalent:

    1. OLSE (Xβ) = BLUE(Xβ).

    2. OLSP (𝜺) = BLUP(𝜺).

    3. PX Σ = Σ PX.

    4. X⊥ Σ = Σ X⊥.

    5. r[X, ΣX] = r(X).

    6. r[X⊥, ΣX⊥]= r(X⊥).

    7. ℛ(ΣX) ⊆ ℛ(X).

    8. ℛ(ΣX⊥) ⊆ ℛ(X⊥).

    9. ℛ(ΣX) = ℛ(Σ) ∩ ℛ(X).

    10. ℛ(ΣX⊥) = ℛ(Σ) ∩ ℛ(X⊥).

Proof

From (37) and (69), the difference of OLSP (ϕ) and BLUP (ϕ) is given by

OLSP(ϕ)−BLUP(ϕ)=(KX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+−U[X,ΣX⊥]⊥)y.

Hence OLSP (ϕ) = BLUP(ϕ) holds if and only if the coefficient matrix of y is null, i.e.,

KX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+−U[X,ΣX⊥]⊥=0.

From Lemma 2.7, the matrix equation is solvable for U if and only if

rKX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+[X,ΣX⊥]⊥=r([X,ΣX⊥]⊥), (86)

where by (13),

rKX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+[X,ΣX⊥]⊥=rKX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+0In[X,ΣX⊥]−r[X,ΣX⊥]=rKX++JX⊥KJΣX⊥InXΣX⊥−r[X,Σ]=r00JΣX⊥−(KX++JX⊥)ΣX⊥In00−r[X,Σ]=n+r[(K−JX)X+ΣX⊥]−r[X,Σ]=n+r[(K−JX)X′X+X′ΣX⊥]−r[X,Σ]=rX′XX′ΣX⊥K−JX0+n−r(X)−r[X,Σ](by(16))=rX′XX′Σ0X′K−JX0+n−2r(X)−r[X,Σ](by(14)), (87)

r([X,ΣX⊥]⊥)=n−r[X,Σ]. (88)

Substituting (87) and (88) into (86) yields the following equalities

(K−JX)X+ΣX⊥=0,rX′XX′ΣX⊥K−JX0=r(X),rX′XX′Σ0X′K−JX0=2r(X),

establishing the equivalences of (i), (ii), (iii), and (iv) in (a). The equivalences of (ii) and (v), (iii) and (vi), (iv) and (vii) in (a) follow from Lemma 2.2(a) and (b). Results (b)–(d) are special cases of (a) for different choices of K and J.

Note that (K − JX)X+ΣX⊥ = 0 in (ii) of Theorem 6.1(a) is a linear matrix equation for K and J to satisfy. Solving this equation will produce allϕ such that OLSP(ϕ) = BLUP(ϕ) hold. Concerning the relationships between D[ϕ − BLUP(ϕ)] and D[ϕ − OLSP(ϕ)], we have the following results.

Theorem 6.2

Assume that ϕ in (2) is predictable under (1), and let BLUP(ϕ) and OLSP(ϕ) be as given in (37) and (69), respectively. Then

i−{D[ϕ−BLUP(ϕ)]−D[ϕ−OLSP(ϕ)]}=r{D[ϕ−BLUP(ϕ)]−D[ϕ−OLSP(ϕ)]}=rX′XX′Σ0X′K−JX0−2r(X). (89)

In particular,

i−{D[BLUE(Kβ)]−D[OLSE(Kβ)]}=r{D[BLUE(Kβ)]−D[OLSE(Kβ)]}=rΣXX0X′X0K′−2r(X)ifKβisestimable, (90)

i−{D[BLUE(Xβ)]−D[OLSE(Xβ)]}=r{D[BLUE(Xβ)]−D[OLSE(Xβ)]}=r[X,ΣX]−r(X)<n, (91)

i−{D[BLUE(β)]−D[OLSE(β)]}=r{D[OLSE(β)]−D[BLUE(β)]}=r[ΣX,X]−p<nifβisestimable, (92)

i−{D[ε−BLUP(ε)]−D[ε−OLSP(ε)]}=r{D[ε−BLUP(ε)]−D[ε−OLSP(ε)]}=r[ X,Σ X]−r( X)<n, (93)

i+{D[BLUP(ε)]−D[OLSP(ε)]}=i−{D[BLUP(ε)]−D[OLSP(ε)]}=r[X,ΣX]−r(X)<n, (94)

r{D[BLUP(ε)]−D[OLSP(ε)]}=2r[X,ΣX]−2r(X)<2n. (95)

In consequence, the following results hold.

  1. D[ϕ−BLUP(ϕ)]≺D[ϕ−OLSP(ϕ)]⇔rX′XX′Σ0X′K−JX0=2r(X)+k.

  2. D[BLUE(Kβ)]≺D[OLSE(Kβ)]⇔rΣXX0X′X0K′=2r(X)+k.

Proof

By (37), (43), (69) and (73)

D[ϕ−BLUP(ϕ)]−D[ϕ−OLSP(ϕ)]=σ2([K,JΣX⊥][X,ΣX⊥]+−J)Σ([K,JΣX⊥][X,ΣX⊥]+−J)′−σ2(KX+−JPX)Σ(KX+−JPX)′, (96)

D[OLSP(ϕ)−BLUP(ϕ)]=σ2(KX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+)Σ(KX++JX⊥−[K,JΣX⊥][X,ΣX⊥]+)′. (97)

Applying (81) to (96), we first obtain

i±(D[ϕ−BLUP(ϕ)]−D[ϕ−OLSP(ϕ)])=i∓σ2X′ΣX0X′X0−D[ϕ−BLUP(ϕ)]K−JXX′XK′−X′J′0−r(X). (98)

Furthermore, substituting D[ϕ − BLUP(ϕ)] in (43) into the block matrix in (98) and applying (61), we obtain

i∓σ2X′ΣX0X′X0−D[ϕ−BLUP(ϕ)]K−JXX′XK′−X′J′0=i∓X′ΣX0X′X0−([K,JΣX⊥][X,ΣX⊥]+−J)Σ([K,JΣX⊥][X,ΣX⊥]+−J)′K−JXX′XK′−X′J′0=i∓X′ΣX0X′X00K−JXX′XK′−X′J′0−0Ik0([K,JΣX⊥][X,ΣX⊥]+−J)Σ([K,JΣX⊥][X,ΣX⊥]+−J)′[0,Ik,0]=i±Σ0X0−X′ΣX0X′X00K−JXX′XK′−X′J′00K−JX0X′[0,K′−X′J′0]0+i∓(X⊥ΣX⊥)−r[X,Σ]=i±Σ000X0−X′ΣX0−X′X0000−K+JXK−JX0−X′X−K′+X′J′00X′0K′−X′J′00+i∓(X⊥ΣX⊥)−r[X,Σ]=i±Σ000X0−X′ΣX0−X′X−X′X000−K+JX00−X′X−K′+X′J′00X′−X′X000+i∓(X⊥ΣX⊥)−r[X,Σ]=i±ΣΣX00XX′Σ00−X′X0000−K+JX00−X′X−K′+X′J′00X′0000+i∓(X⊥ΣX⊥)−r[X,Σ]=i±Σ0ΣXX000X′X0K′−X′J′X′ΣX′X000X′00000K−JX000+i∓(X⊥ΣX⊥)−r[X,Σ]. (99)

Substituting (99) into (98) gives

i±(D[ϕ−BLUP(ϕ)]−D[OLSP(ϕ)−ϕ])=i±Σ0ΣXX000X′X0K′−X′J′X′ΣX′X000X′00000K−JX000+i∓(X⊥ΣX⊥)−r(X)−r[X,Σ].

Hence,

i+(D[ϕ−BLUP(ϕ)]−D[ϕ−OLSP(ϕ)])=i+Σ0ΣXX000X′X0K′−X′J′X′ΣX′X000X′00000K−JX000+i−(X⊥ΣX⊥)−r(X)−r[X,Σ]=rΣ0ΣXX000X′X0K′−X′J′−r(X)−r[X,Σ]=rΣ00X000X′X00−r(X)−r[X,Σ]=0,i−(D[ϕ−BLUP(ϕ)]−D[ϕ−OLSP(ϕ)])=i−Σ0ΣXX000X′X0K′−X′J′X′ΣX′X000X′00000K−JX000+r(X⊥Σ)−r(X)−r[X,Σ]=rX′ΣX′XX′00K−JX−2r(X),

establishing (89). Eqs. (90)–(93) follow directly from (89). Eqs. (94) and (95) were proved in [5]. Applying Lemma 2.1 to (89) and (90) yields (a) and (b).

The following results follow from Theorems 5.3, 6.1, and 6.2.

Corollary 6.3

Assume that ϕ in (2) is predictable under (1), and let BLUP(ϕ) and OLSP(ϕ) be as given in (37) and (69), respectively. Then the following statements are equivalent:

  1. BLUP(ϕ) = OLSP(ϕ), i.e., ϕ − BLUP(ϕ) = ϕ − OLSP(ϕ).

  2. D[ϕ − BLUP(ϕ)] = D[ϕ − OLSP(ϕ)].

  3. rX′XX′Σ0X′K−JX0=2r(X),i.e.,R(K−JX)′0⊆RX′X0ΣXX.

Corollary 6.4

Assume that Kβ is estimable under (1). Then the following statisticalfacts are equivalent:

  1. BLUE(Kβ) = OLSE(Kβ).

  2. D[BLUE(Kβ)] = D[OLSE(Kβ)].

  3. Cov{OLSE(Kβ), y} = Cov{BLUE(Kβ), y}.

  4. Cov{OLSE(Kβ), y} = Cov{BLUE(Kβ), OLSE(Xβ)}.

  5. Cov{OLSE(Kβ), OLSP(𝜺)} = 0.

Corollary 6.5

Under the assumption in (1), the following statistical facts are equivalent:

  1. BLUE(Xβ) = OLSE(Xβ).

  2. BLUP(𝜺) = OLSP(𝜺).

  3. D[BLUE(Xβ)] = D[OLSE(Xβ)].

  4. D[BLUP(𝜺)] = D[OLSP(𝜺)].

  5. D[𝜺 − BLUP(𝜺)] = D[𝜺 − OLSP(𝜺)].

  6. D(y) = D[OLSE(Xβ)] + D[OLSP(𝜺)].

  7. D[𝜺 − OLSP(𝜺)] = D(𝜺) − D[OLSP(𝜺)].

  8. Cov{OLSE(Xβ), y} = Cov{BLUE(Xβ), y}.

  9. Cov{OLSE(Xβ), y} = Cov{BLUE(Xβ), OLSE(Xβ)}.

  10. Cov{OLSE(Xβ), OLSE(Xβ)} = Cov{BLUE(Xβ), y}.

  11. Cov{OLSE(Xβ), OLSE(Xβ)} = Cov{BLUE(Xβ), OLSE(Xβ)}.

  12. Cov{OLSE(Xβ), y} = Cov{y, OLSE(Xβ)}.

  13. Cov{OLSP(𝜺), y} = Cov{y, OLSP(𝜺)}.

  14. Cov{OLSE(Xβ), OLSP(𝜺)} = Cov{OLSP(𝜺), OLSE(Xβ)}.

  15. Cov{OLSE(Xβ), OLSP(𝜺)} + Cov{OLSP(𝜺), OLSE(Xβ)} = 0.

  16. Cov{OLSE(Xβ), OLSP(𝜺)} = 0.

Some of the equivalent facts in Corollaries 6.4 and 6.5 were proved in [5, 58]. Furthermore, many interesting vector and matrix norm equalities for OLSEs to be BLUEs can be established. We shall present them in another paper.

7 Conclusions

We have offered a predominantly theoretical coverage of the statistical predictions and estimations by establishing two groups of standard result on exact algebraic expressions of the BLUPs and OLSPs of all unknown parameters and their fundamental properties under (1).We have also established a variety of exact algebraic formulas for calculating ranks and inertias of the matrices associated with the BLUPs and OLSPs, and have used the formulas to characterize many interesting and valuable equalities and inequalities for the dispersion matrices of the BLUPs and OLSPs under (1). The whole work contains a massive amount of useful results related to the world of GLMs and can serve as a comprehensive description of rank/inertia theory in the parameter prediction and estimation problems under GLMs.

Statistical theory and methods often require various mathematical computations with vectors and matrices. In particular, formulas and algebraic tricks for handling matrices in linear algebra and matrix theory play important roles in the derivations and characterizations of estimators and predictors and their features and performances under linear regression models. So that matrix theory provides a powerful tool set for addressing statistical problems. There is a long list of handbooks on matrix algebra for statistical analysis published since 1960s; see, e.g., [2, 59–74], while various new algebraic methods were developed regularly in matrix mathematics. But it is rarely the case that the algebraic techniques in matrix theory are ready-made to address statistical challenges. This is why the dialogue between matrix theory and statistics benefits both disciplines.

Although the ranks/inertias of matrices are the conceptual foundation in elementary linear algebra and are the most significant finite integers in reflecting intrinsic properties of matrices, it took a long time in the development of mathematics to establish various analytical and valuable formulas for calculating ranks/inertias of matrices and to use the formulas, as demonstrated in the previous sections, in the intuitive and rigorous derivations of matrix equalities and inequalities in the statistical analysis of GLMs. The present author has been devoting to this subject and has proved thousands of matrix rank/inertia formulas since 1980s by using various skillful calculations for partitioned matrices. This work has provided significant advances to general algebraical and statistical methodology, and a state-of-the-art theory on matrix ranks/inertias with applications has been established. We are now able to use matrix rank/inertia formulas to describe many fundamental properties and features of matrices, such as, establishing and simplifying various complicated matrix expressions, deriving matrix equalities and inequalities that involve generalized inverses of matrices, characterizing definiteness and semi-definiteness of symmetric matrices, and solving matrix optimization problems in the Löwner sense. In the past decade, the present author has been working on applications of the matrix rank/inertia methodology in linear regression analysis, while much experience of using various rank/inertia formulas in statistical inference of GLMs has been achieved, and many fundamental and valuable mathematical and statistical features of predictors/estimators under GLMs have been obtained during this approach. Some recent contributions on this topic by the present author and his collaborators are presented in [5–7, 9, 17, 22, 57, 58, 75– 79], which contain a massive amount of useful results related to the world of GLMs. The whole findings in these papers provide significant advances to algebraical methodology in statistical analysis and inference of GLMs, and can merge into the essential part of unified theory of GLMs.

Acknowledgements

The author thanks anonymous referees for their helpful comments and suggestions on an earlier version of this paper.

This work was supported by the National Natural Science Foundation of China (Grant No. 11271384).

References

[1] Markiewicz A., Puntanen S., All about the ⊥ with its applications in the linear statistical models, Open Math., 2015, 13, 33–5010.1515/math-2015-0005Search in Google Scholar

[2] Puntanen S., Styan G.P.H., Isotalo J., Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty, Springer, Berlin Heidelberg, 201110.1007/978-3-642-10473-2Search in Google Scholar

[3] Rao C.R., Mitra S.K., Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971Search in Google Scholar

[4] Tian Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 2010, 433, 263–29610.1016/j.laa.2010.02.018Search in Google Scholar

[5] Tian Y., Some equalities and inequalities for covariance matrices of estimators under linear model, Stat. Papers, 2015, 10.1007/s00362-015-0707-xSearch in Google Scholar

[6] Tian Y., Guo W., On comparison of dispersion matrices of estimators under a constrained linear model, Stat. Methods Appl., 2016, 25, 623–64910.1007/s10260-016-0350-2Search in Google Scholar

[7] Tian Y., Jiang B., Matrix rank/inertia formulas for least-squares solutions with statistical applications, Spec. Matrices, 2016, 4, 130–14010.1515/spma-2016-0013Search in Google Scholar

[8] Tian Y., Jiang B., Quadratic properties of least-squares solutions of linear matrix equations with statistical applications, Comput. Statist., 10.1007/s00180-016-0693-zSearch in Google Scholar

[9] Dong B., Guo W., Tian Y., On relations between BLUEs under two transformed linear models, J. Multivariate Anal., 2014, 131, 279–29210.1016/j.jmva.2014.07.005Search in Google Scholar

[10] Lowerre J.M., Some simplifying results on BLUEs, J. Amer. Stat. Assoc., 1977, 72, 433–43710.1080/01621459.1977.10481015Search in Google Scholar

[11] Rao C.R., A lemma on optimization of matrix function and a review of the unified theory of linear estimation, In: Y. Dodge (ed.), Statistical Data Analysis and Inference, North-Holland, Elsevier, 1989, 397–41710.1016/B978-0-444-88029-1.50042-3Search in Google Scholar

[12] Goldberger A.S., Best linear unbiased prediction in the generalized linear regression models, J. Amer. Stat. Assoc., 1962, 57, 369–37510.1080/01621459.1962.10480665Search in Google Scholar

[13] Marsaglia G., Styan G.P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 1974, 2, 269–29210.1080/03081087408817070Search in Google Scholar

[14] Tian Y., More on maximal and minimal ranks of Schur complements with applications, Appl. Math. Comput., 2004, 152, 675–69210.1016/S0096-3003(03)00585-XSearch in Google Scholar

[15] Penrose R., A generalized inverse for matrices, Proc. Cambridge Phil. Soc., 1955, 51, 406–41310.1017/S0305004100030401Search in Google Scholar

[16] Lange K., Chi E.C., Zhou H., A brief survey of modern optimization for statisticians, Internat. Stat. Rev., 2014, 82, 46–7010.1111/insr.12022Search in Google Scholar

[17] Tian Y., A new derivation of BLUPs under random-effects model, Metrika, 2015, 78, 905–91810.1007/s00184-015-0533-0Search in Google Scholar

[18] Rao C.R., Unified theory of linear estimation, Sankhyā Ser. A, 1971, 33, 371–394Search in Google Scholar

[19] Rao C.R., Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix, J. Multivariate Anal., 1973, 3, 276–29210.1016/0047-259X(73)90042-0Search in Google Scholar

[20] Rao C.R., Toutenburg H., Shalabh, Heumann C., Linear Models and Generalizations Least Squares and Alternatives, 3rd ed., Springer, Berlin Heidelberg, 2008Search in Google Scholar

[21] Searle S.R., The matrix handling of BLUE and BLUP in the mixed linear model, Linear Algebra Appl., 1997, 264, 291–31110.1016/S0024-3795(96)00400-4Search in Google Scholar

[22] Gan S., Sun Y., Tian Y., Equivalence of predictors under real and over-parameterized linear models, Comm. Stat. Theory Meth., 2016, 10.1080/03610926.2015.1100742Search in Google Scholar

[23] Tian Y., Jiang, B., A new analysis of the relationships between a general linear model and its mis-specified forms, J. Korean Stat. Soc., 2016, 10.1016/j.jkss.2016.08.004Search in Google Scholar

[24] Baksalary J.K., Puntanen S., Characterizations of the best linear unbiased estimator in the general Gauss–Markov model with the use of matrix partial orderings, Linear Algebra Appl., 1990, 127, 363–37010.1016/0024-3795(90)90349-HSearch in Google Scholar

[25] Baksalary J.K., Puntanen S., Styan G.P.H., A property of the dispersion matrix of the best linear unbiased estimator in the general Gauss–Markov model, Sankhyā Ser. A, 1990, 52, 279–296Search in Google Scholar

[26] Isotalo J., Puntanen S., Styan G.P.H., The BLUE’s covariance matrix revisited: A review, J. Stat. Plann. Inference, 2008, 138, 2722–273710.1016/j.jspi.2008.03.029Search in Google Scholar

[27] Puntanen S., Styan G.P.H., Tian Y., Three rank formulas associated with the covariance matrices of the BLUE and the OLSE in the general linear model, Econometric Theory, 2005, 21, 659–66410.1017/S0266466605050292Search in Google Scholar

[28] Abdulle A., Wanner G., 200 years of least squares method, Elem. Math., 2002, 57, 45–6010.1007/PL00000559Search in Google Scholar

[29] Farebrother R.W., Some early statistical contributions to the theory and practice of linear algebra, Linear Algebra Appl., 1996, 237/238, 205–22410.1016/0024-3795(95)00600-1Search in Google Scholar

[30] Paris Q., The dual of the least-squares method, Open J. Statist., 2015, 5, 658–66410.4236/ojs.2015.57067Search in Google Scholar

[31] Stigler S.M., Gauss and the invention of least squares, Ann. Stat., 1981, 9, 465–47410.2307/j.ctv1pdrpsj.21Search in Google Scholar

[32] Graybill F.A., An Introduction to Linear Statistical Models, Vol. I, McGraw–Hill, New York, 1961Search in Google Scholar

[33] Searle S.R., Linear Models, Wiley, New York, 1971Search in Google Scholar

[34] Puntanen S., Styan G.P.H., The equality of the ordinary least squares estimator and the best linear unbiased estimator, with comments by O. Kempthorne, S.R. Searle, and a reply by the authors, Amer. Statistican, 1989, 43, 153–16410.2307/2685065Search in Google Scholar

[35] Alalouf I.S., Styan G.P.H., Characterizations of the conditions for the ordinary least squares estimator to be best linear unbiased, in: Y.P. Chaubey, T.D. Dwivedi (Eds.), Topics in Applied Statistics, Concordia University, Montréal, 1984, 331–344Search in Google Scholar

[36] Baksalary J.K., Criteria for the equality between ordinary least squares and best linear unbiased estimators under certain linear models, Canad. J. Stat., 1988, 16, 97–10210.2307/3315067Search in Google Scholar

[37] Baksalary J.K., Kala R., An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator, J. Stat. Plann. Inference, 1977, 1, 309–31210.1016/0378-3758(77)90015-5Search in Google Scholar

[38] Baksalary J.K., Kala R., Simple least squares estimation versus best linear unbiased prediction, J. Stat. Plann. Inference, 1981, 5, 147–15110.1016/0378-3758(81)90024-0Search in Google Scholar

[39] Baksalary J.K., van Eijnsbergen A.C., Comparison of two criteria for ordinary-least-squares estimators to be best linear unbiased estimators, Amer. Statistican, 1988, 42, 205–20810.1080/00031305.1988.10475567Search in Google Scholar

[40] Baksalary O.M., Trenkler G., Between OLSE and BLUE, Aust. N.Z.J. Stat., 2011, 53, 289–30310.1111/j.1467-842X.2011.00635.xSearch in Google Scholar

[41] Baksalary O.M., Trenkler G., Liski E.P., Let us do the twist again, Stat. Papers, 2013, 54, 1109–111910.1007/s00362-013-0512-3Search in Google Scholar

[42] Haslett S.J., Isotalo J., Liu Y., Puntanen S., Equalities between OLSE, BLUE and BLUP in the linear model, Stat. Papers, 2014, 55, 543–56110.1007/s00362-013-0500-7Search in Google Scholar

[43] Haslett S.J., Puntanen S., A note on the equality of the BLUPs for new observations under two linear models, Acta Comment. Univ. Tartu. Math., 2010, 14 27–3310.12697/ACUTM.2010.14.03Search in Google Scholar

[44] Haslett S.J., Puntanen S., Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Stat. Papers, 2010, 51, 465–47510.1007/s00362-009-0219-7Search in Google Scholar

[45] Haslett S.J., Puntanen S., On the equality of the BLUPs under two linear mixed models, Metrika, 2011, 74, 381–39510.1007/s00184-010-0308-6Search in Google Scholar

[46] Herzberg A.M., Aleong J., Further conditions on the equivalence of ordinary least squares and weighted least squares estimators with examples, in: J. Lanke, G. Lindgren (Eds.), Contributions to Probability and Statistics in Honour of Gunnar Blom, University of Lund, 1985, 127–142Search in Google Scholar

[47] Isotalo J., Puntanen S., A note on the equality of the OLSE and the BLUE of the parametric functions in the general Gauss–Markov model, Stat. Papers, 2009, 50, 185–19310.1007/s00362-007-0055-6Search in Google Scholar

[48] Jiang B., Sun Y., On the equality of estimators under a general partitioned linear model with parameter restrictions, Stat. Papers, 2016, 10.1007/S00362-016-0837-9Search in Google Scholar

[49] Kruskal W., When are Gauss–Markov and least squares estimators identical? A coordinate-free approach, Ann. Math. Statist., 1968, 39, 70–7510.1214/aoms/1177698505Search in Google Scholar

[50] Liski E.P., Puntanen S., Wang S., Bounds for the trace of the difference of the covariance matrices of the OLSE and BLUE, Linear Algebra Appl., 1992, 176, 121–13010.1016/0024-3795(92)90214-USearch in Google Scholar

[51] McElroy F.W., A necessary and sufficient condition that ordinary least-squares estimators be best linear unbiased, J. Amer. Stat. Assoc., 1967, 62, 1302–130410.1080/01621459.1967.10500935Search in Google Scholar

[52] Milliken G.A., Albohali M., On necessary and sufficient conditions for ordinary least squares estimators to be best linear unbiased estimators, Amer. Statistican, 1984, 38, 298–29910.1080/00031305.1984.10483234Search in Google Scholar

[53] Norlèn U., The covariance matrices for which least squares is best linear unbiased, Scand. J. Statist., 1975, 2, 85–90Search in Google Scholar

[54] Styan G.P.H., When does least squares give the best linear unbiased estimate?, in: D.G. Kabe, R.P. Gupta (Eds.), Multivariate Statistical Inference, North-Holland, Amsterdam, 1973, 241–246Search in Google Scholar

[55] Tian Y., On equalities of estimations of parametric functions under a general linear model and its restricted models, Metrika, 2010, 72, 313–33010.1007/s00184-009-0255-2Search in Google Scholar

[56] Tian Y., On properties of BLUEs under general linear regression models, J. Stat. Plann. Inference, 2013, 143, 771–78210.1016/j.jspi.2012.10.005Search in Google Scholar

[57] Tian Y., Zhang J., Some equalities for estimations of partial coefficients under a general linear regression model, Stat. Papers, 2011, 52, 911–92010.1007/s00362-009-0298-5Search in Google Scholar

[58] Tian Y., Zhang X., On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model, Stat. Prob. Lett., 2016, 112, 105–11210.1016/j.spl.2016.01.019Search in Google Scholar

[59] Abadir K.M., Magnus J.R., Matrix Algebra, Cambridge University Press, 200510.1017/CBO9780511810800Search in Google Scholar

[60] Banerjee S., Roy A., Linear Algebra and Matrix Analysis for Statistics, CRC Press, New York, 201410.1201/b17040Search in Google Scholar

[61] Bapat R.B., Linear Algebra and Linear Models, 3rd ed., Springer, Berlin Heidelberg, 201210.1007/978-1-4471-2739-0Search in Google Scholar

[62] Eldén L., Matrix Methods in Data Mining and Pattern Recognition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 200710.1137/1.9780898718867Search in Google Scholar

[63] Fieller N., Basics of Matrix Algebra for Statistics with R, Chapman and Hall/CRC, 2015Search in Google Scholar

[64] Gentle J.E., Numerical Linear Algebra for Applications in Statistics, Springer, Berlin Heidelberg, 199810.1007/978-1-4612-0623-1Search in Google Scholar

[65] Gentle J.E., Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer, Berlin Heidelberg, 200710.1007/978-0-387-70873-7Search in Google Scholar

[66] Graybill F.A., Matrices with Applications in Statistics, 2nd ed., Brooks/Cole, 2002Search in Google Scholar

[67] Harville D.A., Matrix Algebra From a Statistician’s Perspective, Springer, New York, 199710.1007/b98818Search in Google Scholar

[68] Harville D.A., Matrix Algebra: Exercises and Solutions, Springer, New York, 200110.1007/978-1-4613-0181-3Search in Google Scholar

[69] Healy M.J.R., Matrices for Statistics, 2nd ed., Oxford University Press, 2000Search in Google Scholar

[70] Magnus J.R., Neudecker H., Matrix Differential Calculus with Applications in Statistics and Econometrics, Revised edition of the 1988 original, Wiley, New York, 199910.2307/2531754Search in Google Scholar

[71] Rao C.R., Rao M.B., Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore, 199810.1142/3599Search in Google Scholar

[72] Schott J.R., Matrix Analysis for Statistics, 2nd ed., Wiley, Hoboken, NJ, 2005Search in Google Scholar

[73] Searle S.R., Matrix Algebra Useful for Statistics, Wiley, New York, 1982Search in Google Scholar

[74] Seber G.A.F., A Matrix Handbook for Statisticians, Wiley, New York, 200810.1002/9780470226797Search in Google Scholar

[75] Lu C., Gan S., Tian Y., Some remarks on general linear model with new regressors, Stat. Prob. Lett., 2015, 97, 16–2410.1016/j.spl.2014.10.015Search in Google Scholar

[76] Tian Y., A matrix handling of predictions under a general linear random-effects model with new observations, Electron. J. Linear Algebra, 2015, 29, 30–4510.13001/1081-3810.2895Search in Google Scholar

[77] Tian Y., Jiang B., Equalities for estimators of partial parameters under linear model with restrictions, J. Multivariate Anal., 2016, 143, 299–31310.1016/j.jmva.2015.09.007Search in Google Scholar

[78] Tian Y., Jiang B., An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices, Linear Multilinear Algebra, 2016, 64, 2351–236710.1080/03081087.2016.1155533Search in Google Scholar

[79] Zhang X., Tian Y., On decompositions of BLUEs under a partitioned linear model with restrictions, Stat. Papers, 2016, 57, 345–36410.1007/s00362-014-0654-ySearch in Google Scholar
Received: 2016-3-15
Accepted: 2017-1-3
Published Online: 2017-2-27

© 2017 Tian

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
https://doi.org/10.1515/math-2017-0013
Keywords for this article
General linear model; Predictor; Estimator; Covariance matrix; Rank; Inertia; Equality; Inequality
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2017-0013/html
Scroll to top button