Displaying similar documents to “All about the ⊥ with its applications in the linear statistical models”

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

Yongge 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...

Possible numbers ofx’s in an {x,y}-matrix with a given rank

Chao Ma (2017)

Open Mathematics

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Let x, y be two distinct real numbers. An {x, y}-matrix is a matrix whose entries are either x or y. We determine the possible numbers of x’s in an {x, y}-matrix with a given rank. Our proof is constructive.

Remarks on the Sherman-Morrison-Woodbury formulae

Miroslav Fiedler (2003)

Mathematica Bohemica

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We present some results on generalized inverses and their application to generalizations of the Sherman-Morrison-Woodbury-type formulae.

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

Yongge Tian, Bo Jiang (2016)

Special Matrices

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Least-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...

On the Yang-Baxter-like matrix equation for rank-two matrices

Duanmei Zhou, Guoliang Chen, Jiu Ding (2017)

Open Mathematics

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Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

Bias of LS estimators in nonlinear regression models with constraints. Part II: Biadditive models

Jean-Baptiste Denis, Andrej Pázman (1999)

Applications of Mathematics

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General results giving approximate bias for nonlinear models with constrained parameters are applied to bilinear models in anova framework, called biadditive models. Known results on the information matrix and the asymptotic variance matrix of the parameters are summarized, and the Jacobians and Hessians of the response and of the constraints are derived. These intermediate results are the basis for any subsequent second order study of the model. Despite the large number of parameters...