Linear Models and the Relevant Distributions and Matrix Algebra: 1st Edition (Hardback) book cover

Linear Models and the Relevant Distributions and Matrix Algebra

1st Edition

By David A. Harville

Chapman and Hall/CRC

524 pages

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pub: 2018-03-13
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Description

Linear Models and the Relevant Distributions and Matrix Algebra provides in-depth and detailed coverage of the use of linear statistical models as a basis for parametric and predictive inference. It can be a valuable reference, a primary or secondary text in a graduate-level course on linear models, or a resource used (in a course on mathematical statistics) to illustrate various theoretical concepts in the context of a relatively complex setting of great practical importance.

Features:

  • Provides coverage of matrix algebra that is extensive and relatively self-contained and does so in a meaningful context
  • Provides thorough coverage of the relevant statistical distributions, including spherically and elliptically symmetric distributions
  • Includes extensive coverage of multiple-comparison procedures (and of simultaneous confidence intervals), including procedures for controlling the k-FWER and the FDR
  • Provides thorough coverage (complete with detailed and highly accessible proofs) of results on the properties of various linear-model procedures, including those of least squares estimators and those of the F test.
  • Features the use of real data sets for illustrative purposes
  • Includes many exercises

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.

Reviews

"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

Table of Contents

Preface

1 Introduction

Linear Statistical Models

Regression Models

Classificatory Models

Hierarchical Models and Random-EffectsModels

Statistical Inference

An Overview

2 Matrix Algebra: a Primer

The Basics

Partitioned Matrices and Vectors

Trace of a (Square) Matrix

Linear Spaces

Inverse Matrices

Ranks and Inverses of Partitioned Matrices

OrthogonalMatrices

IdempotentMatrices

Linear Systems

Generalized Inverses

Linear Systems Revisited

Projection Matrices

Quadratic Forms

Determinants

Exercises

Bibliographic and Supplementary Notes

3 Random Vectors and Matrices

Expected Values

Variances, Covariances, and Correlations

Standardized Version of a Random Variable

Conditional Expected Values and Conditional Variances and Covariances

Multivariate Normal Distribution

Exercises

Bibliographic and Supplementary Notes

4 The General Linear Model

Some Basic Types of Linear Models

Some Specific Types of Gauss-Markov Models (With Examples)

Regression

Heteroscedastic and Correlated Residual Effects

Multivariate Data

vi Contents

Exercises

Bibliographic and Supplementary Notes

5 Estimation and Prediction: Classical Approach

Linearity and Unbiasedness

Translation Equivariance

Estimability

The Method of Least Squares

Best LinearUnbiased or Translation-EquivariantEstimation of Estimable Functions

(Under the G-M Model)

Simultaneous Estimation

Estimation of Variability and Covariability

Best (Minimum-Variance) Unbiased Estimation

Likelihood-Based Methods

Prediction

Exercises

Bibliographic and Supplementary Notes

6 Some Relevant Distributions and Their Properties

Chi-Square, Gamma, Beta, and Dirichlet Distributions

Noncentral Chi-Square Distribution

Central and Noncentral F Distributions

Central, Noncentral, and Multivariate t Distributions

Moment Generating Function of the Distribution of One or More Quadratic Forms

or Second-Degree Polynomials (in a Normally Distributed Random Vector)

Distribution of Quadratic Forms or Second-Degree Polynomials (in a Normally

Distributed Random Vector): Chi-Squareness

The Spectral Decomposition, With Application to the Distribution of Quadratic

Forms

More on the Distribution of Quadratic Forms or Second-Degree Polynomials (in a

Normally Distributed Random Vector)

Exercises

Bibliographic and Supplementary Notes

7 Confidence Intervals (or Sets) and Tests of Hypotheses

"Setting the Stage": Response Surfaces in the Context of a Specific Application and

in General

Augmented G-M Model

The F Test (and Corresponding Confidence Set) and the S Method

Some Optimality Properties

One-Sided t Tests and the Corresponding Confidence Bounds

The Residual Variance : Confidence Intervals and Tests

Multiple Comparisons and Simultaneous Confidence Intervals: Some Enhancements

Prediction

Exercises

Bibliographic and Supplementary Notes

References

Index

About the Author

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.

About the Series

Chapman & Hall/CRC Texts in Statistical Science

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis