Some equalities and inequalities for covariance matrices of estimators under linear model

Tian, Yongge

doi:10.1007/s00362-015-0707-x

Cited by 20 publications

(6 citation statements)

References 20 publications

(14 reference statements)

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“…/ in (43) into the block matrix in (98) and applying (61), we obtain 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.…”

Section: Twists Of Blups and Olsps Under Glmsmentioning

confidence: 73%

“…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 DOE BLUP.…”

Section: Rank/inertia Formulas For Dispersion Matrices Of Blupsmentioning

confidence: 99%

“…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][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

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

Tian

2017

Open Mathematics

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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.

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Section: Twists Of Blups and Olsps Under Glmsmentioning

confidence: 73%

Section: Rank/inertia Formulas For Dispersion Matrices Of Blupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Tian

2017

Open Mathematics

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“…We experience the least-norm to the system (7) in this section. By the definition and [55], we can get the following result easily.…”

Section: A New Expression Of the General Solution To The Systemmentioning

confidence: 97%

Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

Rehman¹,

Kyrchei²

2023

Inverse Problems - Recent Advances and Applications

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Keeping in view that a lot of physical systems with inverse problems can be written by matrix equations, the least-norm of the solution to a general Sylvester matrix equation with restrictions A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=Cc, is researched in this chapter. A novel expression of the general solution to this system is established and necessary and sufficient conditions for its existence are constituted. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore–Penrose inverse previously obtained by one of the authors.

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“…in [11][12][13]. Some recent work on the MRM in the analysis of additive decompositions of BLUEs under linear models were presented in [4][5][6], while some contributions on MRM in the statistical analysis of CGLMs can be found in [14][15][16][17][18][19][20][21][22][23][24].…”

Section: Some Preliminaries In Linear Algebramentioning

confidence: 99%

On decompositions of estimators under a general linear model with partial parameter restrictions

2017

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A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a general linear model with partial parameter restrictions. We will demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear model (CGLM) as the sums of estimators under submodels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of linear regression models.

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Some equalities and inequalities for covariance matrices of estimators under linear model

Cited by 20 publications

References 20 publications

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

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

Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

On decompositions of estimators under a general linear model with partial parameter restrictions

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