Linear models. An integrated approach (Q2713469)

This is a new and comprehensive book on linear models mainly based on the matrix-approach, not the coordinate-free approach. The authors call it an integrated one since they use two techniques for dealing with the subject. The first technique is the linear Lehmann-Scheffé theorem and using LZFs (linear zero functions). This idea goes probably back to the late Suijt Kumar Mitra from the Indian Statistical Institute. The second technique is covariance-adjustment which is essentially described by the conditional covariance-matrix in the normal case and the Schur-complement. The book consists of 11 chapters and a lot of exercises for every chapter. The solutions of the exercises can be found at the end of the book. The bibliography is comprehensive but not exhaustive. Nevertheless, the references are in general up to-date.NEWLINENEWLINE The first chapter introduces the linear model, the second one gives the necessary results in matrix theory and linear algebra. The third chapter reviews basic concepts and results from mathematical statistics. Chapter 4, on the estimation in the general Gauss-Markov model, is the central chapter of this monograph. It is followed by chapter 5 on confidence intervals, hypotheses testing and prediction in linear models. Chapter 6 is on applications, mainly in variance analysis. The linear model with possibly singular covariance-matrix is the subject of chapter 7. The BLUE is obtained by rewriting the linear model as a restriction model. The BLUE can in this case be obtained without inverting the covariance-matrix. This approach is already contained in the reviewer's monograph, The coordinate-free approach to Gauss-Markov estimation. (1970; Zbl 0215.26504). The 8\,th chapter is entitled ``Misspecification or unknown dispersion''. Here also a part of the theory of variance components is dealt with, but Jordan algebras and quadratic subspaces do not appear. Chapter 9 is called ``Updates in the general linear model''. It deals with adding and removing observations and/or variables, including regression diagnostics and the Kalman filter. This chapter is followed by a chapter on the Multivariate Linear Regression Model and a chapter on Linear Inference. It includes such topics as linear sufficiency, linear Bayes-estimation, linear minimax estimation, and miscellenious subjects.NEWLINENEWLINE This monograph can highly be recommended to anyone who is interested in an up to date information on linear models.