scholarly journals Matrix rank/inertia formulas for least-squares solutions with statistical applications

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Yongge Tian ◽  
Bo Jiang
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Linear Models ◽  
Ordinary Least Squares ◽  
General Linear ◽  
Linear Matrix ◽  
Unknown Parameters ◽  
Computational Mathematics ◽  
Linear Matrix Equation ◽  
Quadratic Matrix

AbstractLeast-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z) subject to the LSS of AZ = B, and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.

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

2017 ◽  
Vol 15 (1) ◽  
pp. 126-150 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Statistical Analysis ◽  
Least Squares ◽  
Linear Models ◽  
Linear Regression Analysis ◽  
Ordinary Least Squares ◽  
Covariance Matrices ◽  
General Linear ◽  
General Linear Models ◽  
Best Linear Unbiased ◽  
Best Linear Unbiased Estimators

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.

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

Least squares solutions with special structure to the linear matrix equation AXB=C

2011 ◽  
Vol 217 (24) ◽  
pp. 10049-10057 ◽  
Author(s):  
Fengxia Zhang ◽  
Ying Li ◽  
Wenbin Guo ◽  
Jianli Zhao
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Special Structure ◽  
Linear Matrix ◽  
Linear Matrix Equation

scholarly journals Sensitivity analysis in linear models

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Shuangzhe Liu ◽  
Tiefeng Ma ◽  
Yonghui Liu
Keyword(s):  
Sensitivity Analysis ◽  
Least Squares ◽  
Linear Model ◽  
Linear Models ◽  
General Linear Model ◽  
Ordinary Least Squares ◽  
The Other ◽  
Regression Coefficients ◽  
General Linear ◽  
Least Squares Estimators

AbstractIn this work, we consider the general linear model or its variants with the ordinary least squares, generalised least squares or restricted least squares estimators of the regression coefficients and variance. We propose a newly unified set of definitions for local sensitivity for both situations, one for the estimators of the regression coefficients, and the other for the estimators of the variance. Based on these definitions, we present the estimators’ sensitivity results.We include brief remarks on possible links of these definitions and sensitivity results to local influence and other existing results.

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