Linear Models and the Relevant Distributions and Matrix Algebra (Hardcover)

Preface

1 Introduction

Linear Statistical Models

Regression Models

Classificatory Models

Hierarchical Models and Random-EffectsModels

Statistical Inference

An Overview

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

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

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

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

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

"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