Matrix Tricks for Linear Statistical Models

Puntanen, Simo; Styan, George P. H.; Isotalo, Jarkko

doi:10.1007/978-3-642-10473-2

Cited by 122 publications

(64 citation statements)

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“…Comparing Corollary 3 to Proposition 17.5 in [6] (for square matrices), we can see that Corollary 3 gives us an extension of that version of the Cochran's Theorem without assuming A + B = I. Indeed, for squares matrices A and B of the same size, it is easy to see that A + B = I implies N (A) ⊆ R(B).…”

Section: Sylvester's Inequality and Equalitymentioning

confidence: 80%

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

Thome

2016

Aequat. Math.

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Thome, N. (2016). Inequalities and equalities for l = 2 (Sylvester), l = 3 (Frobenius), and l > 3 matrices. Aequationes Mathematicae. 90 (5) AbstractThis paper gives simple proofs of Sylvester ( = 2) and Frobenius ( = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for > 3 matrices and an application to generalized inverses are provided.

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Section: Sylvester's Inequality and Equalitymentioning

confidence: 80%

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

Thome

2016

Aequat. Math.

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“…According to Rao [1], the problem of inference from a linear model can be completely solved when one has obtained an arbitrary generalized inverse of the partitioned matrix Z . This approach based on the numerical evaluation of an inverse of the partitioned matrix Z is known as the IPM method, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section:  mentioning

confidence: 99%

IPM yöntemi ile alt parametrelerin tahmini

Eriş¹,

Güler

2016

SAUFenBilDer

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In this study, a general partitioned linear modelis considered to determine the best linear unbiased estimators (BLUEs) of subparameters 1 1 X  and 2 2 X  . Some results are given related to the BLUEs of subparameters by using the inverse partitioned matrix (IPM) method based on a generalized inverse of a symmetric block partitioned matrix which is obtained from the fundamental BLUE equation.

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“…D I m AA C , and F A D I n A C A. All about the orthogonal projectors P A , E A , and F A with their applications in the linear statistical models can be found in [1][2][3]. The symbols i C .A/ and i .A/ for A D A 0 2 R m 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.…”

Section: Introductionmentioning

confidence: 99%

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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Matrix Tricks for Linear Statistical Models

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 122 publications

References 1 publication

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

IPM yöntemi ile alt parametrelerin tahmini

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

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