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The book presents procedures for making statistical inferences on the basis of the classical linear statistical model, and discusses the various properties of those procedures. Supporting material on matrix algebra and statistical distributions is interspersed with a discussion of relevant inferential procedures and their properties. The coverage ranges from MS-level to advanced researcher. In particular, the material in chapters 6-7 is not covered in an approachable manner in any other books, and greatly generalizes the traditional normal-based linear regression model to the elliptical distributions, thus greatly elucidating the advanced reader on just how far this class of models can be extended. Refreshingly, the material also goes beyond the classical 20th century coverage to include some 21st century topics like microarray (big) data analysis, and control of false discovery rates in large scale experiments...From the point of view of an advanced instructor and researcher on the subject, I very strongly recommend publication...Note that...this book provides the coverage of 3 books, hence the title purporting to provide a 'unified approach' (of 3 related subjects) is indeed accurate. ~Alex Trindade, Texas Tech University The book is very well written, with exceptional attention to details. It provides detailed derivations or proofs of almost all the results, and offers in-depth coverage of the topics discussed. Some of these materials (e.g., spherical/elliptical distributions) are hard to find from other sources. Anyone who is interested in linear models should benefit from reading this book and find it especially useful for a thorough understanding of the linear-model theory in a unified framework... The book is a delight to read. ~Huaiqing Wu, Iowa State University This book is useful in two ways: an excellent text book for a graduate level linear models course, and for those who want to learn linear models from a theoretical perspective...I genuinely enjoyed reading Ch 1and Ch 4 (Introduction and General Linear Models). Often, the hardest part of teaching linear models from a theoretical perspective is to motivate the students about the utility and generality of such models and the related theory. This book does an excellent job in this area, while presenting a solid theoretical foundation. ~Arnab Maity, North Carolina State University . . . the book does a good job of providing background tools of matrix algebra and distribution theory, basic concepts and advanced level theoretical developments of general linear models in a remarkable way and can be recommended both as a textbook to advanced level graduate students and as a reference book to researchers working on theoretical aspects of general linear models and their applications. ~Anoop Chaturvedi, University of Allahabad One of Harville's major contributions is that this monograph covers both the requisite linear algebra and the statistical theory in a very thorough and balanced manner. It provides a one-stop source of both the statistical and algebraic information needed for a deep understanding of the linear statistical model. In addition, of course, the large range of tools that are introduced and described carefully are invaluable in a many other statistical settings. For these reasons, it has to be compared with some stellar competitors. The seminal books by Rao (1965) and Searle (1971) immediately come to mind. In this reviewer's opinion, Linear Models and the Relevant Distributions and Matrix Algebra, compares with these gems most favourably... In summary, (this) is a first-class volume that will serve as an essential reference for graduate students and established researchers alike in statistics and other related disciplines such as econometrics, biometrics, and psychometrics. As the author discusses, it can also serve as the basis for graduate-level courses which have various emphases. I recommend it strongly. Sometimes you read a book, and you think: 'I wish I had the talent to have written this.' This is definitely one of those books. ~Statistical Papers In summary the book does a good job of providing background tools of matrix algebra and distribution theory, basic concepts and advanced level theoretical developments of general linear models in a remarkable way and can be recommended both as a textbook to advanced level graduate students and as a reference book to researchers working on theoretical aspects of general linear models and their applications. ~Royal Statistical Society Chapter 1 presents an introduction to the fixed and random effects linear model and gives an overview of the issues addressed in this book. Chapter 2 provides all necessary results of matrix algebra for understanding linear model theory. Among them, one finds results about matrix operations, partitioned matrices, traces, linear spaces and ranks of matrices, matrix inversion and generalized inversion, projection and idempotent matri-ces, linear systems, quadratic forms and matrix determinants. Chapter 3 shows results on random vectors and matrices. The results concern means, variances and covariances between random vectors and matrices and their linear combinations. Chapter 3 also provides useful results on the multivariate normal distribution. Chapter 4 discusses lin-ear models examples under three different assumptions about the covariance matrix of observations: homoscedastic independent data, heteroscedastic independent data and heteroscedastic dependent data. The latter includes interesting special cases such as de-pendence of variance and covariance on explanatory (observed) variables, compound symmetry and other variance and covariance forms used in longitudinal data modeling. Of particular interest are spatial data and multivariate data with covariance matrices of blocked form. In Chapter 5, notions like translation equivariance, estimability and identifiability are discussed. Then, the method of least squares is analytically presented along with Best Linear Unbiased Estimation of estimable functions. Of particular in-terest is the variance and covariance estimation section which is based on properties of quadratic forms. Next, maximum likelihood estimation and restricted maximum likeli-hood estimation are discussed along with estimation based on elliptical distributions. Finally, general results on prediction are presented without the multivariate normal-ity assumption. Chapter 6 presents relevant distributions for the study of the linear model. Apart from gamma and Dirichlet distributions, results are presented for central and non-central 2, F and t. Of particular interest are results about the distribution of quadratic forms of multivariate normal random vectors and about their statistical in-dependence. Chapter 7 extends the discussion of Chapter 6 to simultaneous confidence intervals and multiple comparisons. The book is very well written and covers in great detail the theory of linear models, accentuating the relevant topics as the basis for parametric and predictive inference. I found particularly interesting Sections 6.5-6.8 which present results on the distribution of a quadratic form under normality or under spherically or elliptically distributed random vectors. Further, much of the material covered in Chapter 8 is not easily found in just one textbook. Among the important features of the book are: It covers an extensive part of matrix algebra results. It covers all relevant statistical distributions including results on spherically and elliptically symmetric distributions. It presents simultaneous confidence intervals and multiple comparison procedures. It presents many illustrative examples and exercises. As a final comment, the book can be either a reference book or an excellent text book for a graduate level course on linear models or a supplementary material book illustrating various theoretical concepts in the context of multivariate linear model analysis. ~Vassilis G. S. Vasdekis - Mathematical Reviews Clippings - May 2019 This book presents procedures for making statistical inferences on the basis of the classical linear statistical model, and discusses the various properties of those procedures...It could easily form the basis of the (typically required) linear models course taught in traditional statistics MS and PhD-level programs. Moreover, it is written in a way that the easier (MS-level) material is presented early on in the chapters (or at least it's easy to find), with the harder PhD-level material following...All in all, this is a very well-written book that provides an invaluable (and one is tempted to say, definitive ) treatment of this classical subject. ~JASA

The book presents procedures for making statistical inferences on the basis of the classical linear statistical model, and discusses the various properties of those procedures. Supporting material on matrix algebra and statistical distributions is interspersed with a discussion of relevant inferential procedures and their properties. The coverage ranges from MS-level to advanced researcher. In particular, the material in chapters 6-7 is not covered in an approachable manner in any other books, and greatly generalizes the traditional normal-based linear regression model to the elliptical distributions, thus greatly elucidating the advanced reader on just how far this class of models can be extended. Refreshingly, the material also goes beyond the classical 20th century coverage to include some 21st century topics like microarray (big) data analysis, and control of false discovery rates in large scale experiments...From the point of view of an advanced instructor and researcher on the subject, I very strongly recommend publication...Note that...this book provides the coverage of 3 books, hence the title purporting to provide a 'unified approach' (of 3 related subjects) is indeed accurate. ~Alex Trindade, Texas Tech University The book is very well written, with exceptional attention to details. It provides detailed derivations or proofs of almost all the results, and offers in-depth coverage of the topics discussed. Some of these materials (e.g., spherical/elliptical distributions) are hard to find from other sources. Anyone who is interested in linear models should benefit from reading this book and find it especially useful for a thorough understanding of the linear-model theory in a unified framework... The book is a delight to read. ~Huaiqing Wu, Iowa State University This book is useful in two ways: an excellent text book for a graduate level linear models course, and for those who want to learn linear mod

Author Information

David Harville served for 10 years as a mathematical statistician in the Applied Mathematics Research Laboratory of the Aerospace Research Laboratories at Wright-Patterson AFB, Ohio, 20 years as a full professor in Iowa State University’s Department of Statistics where he now has emeritus status, and seven years as a research staff member of the Mathematical Sciences Department of IBM’s T.J. Watson Research Center. He has considerable relevant experience, having taught M.S. and Ph.D. level courses in linear models, been the thesis advisor of 10 Ph.D. graduates, and authored or co-authored two books and more than 80 research articles. His work has been recognized through his election as a Fellow of the American Statistical Association and of the Institute of Mathematical Statistics and as a member of the International Statistical Institute.

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