A First Course in Linear Model Theory

by Ravishanker; Nalini

• ISBN13: 9781584882473
• ISBN10: 1584882476
• Format: Hardcover
• Copyright: 2001-12-21
• Publisher: Chapman & Hall/

Summary

This innovative, intermediate-level statistics text fills an important gap by presenting the theory of linear statistical models at a level appropriate for senior undergraduate or first-year graduate students. The authors' approach introduces readers to the mathematical and statistical concepts and tools, from matrix algebra, linear algebra, and distribution theory that form the backbone for studying the theory and applications of linear models, both univariate and multivariate. It also leads readers through the invaluable first step in their studies of advanced linear models, generalized linear models, nonlinear models, and dynamic models.

Author Biography

Nalini Ravishanker is an Associate Professor and Undergraduate Director in the Department of Statistics at the University of Connecticut, Storrs, USA Dipak K. Dey is a Professor and Head of the Department of Statistics at the University of Connecticut, Storrs, USA

Table of Contents

A Review of Vector and Matrix Algebra	1	(32)
Notation	1	(2)
Basic definitions and properties	3	(30)
Exercises	28	(5)
Properties of Special Matrices	33	(40)
Partitioned matrices	33	(7)
Algorithms for matrix factorization	40	(5)
Symmetric and idempotent matrices	45	(6)
Nonnegative definite quadratic forms and matrices	51	(6)
Simultaneous diagonalization of matrices	57	(1)
Geometrical perspectives	58	(5)
Vector and matrix differentiation	63	(3)
Special operations on matrices	66	(3)
Linear optimization	69	(4)
Exercises	70	(3)
Generalized Inverses and Solutions to Linear Systems	73	(18)
Generalized inverses	73	(9)
Solutions to linear systems	82	(9)
Exercises	88	(3)
The General Linear Model	91	(46)
Model definition and examples	91	(5)
The least squares approach	96	(17)
Estimable functions	113	(5)
Gauss-Markov theorem	118	(4)
Generalized least squares	122	(7)
Estimation subject to linear restrictions	129	(8)
Method of Lagrangian multipliers	129	(2)
Method of orthogonal projections	131	(2)
Exercises	133	(4)
Multivariate Normal and Related Distributions	137	(58)
Multivariate probability distributions	137	(8)
Multivariate normal distribution and properties	145	(19)
Some noncentral distributions	164	(8)
Distributions of quadratic forms	172	(9)
Alternatives to the multivariate normal distribution	181	(14)
Mixture of normals distribution	181	(3)
Spherical distributions	184	(1)
Elliptical distributions	185	(5)
Exercises	190	(5)
Sampling from the Multivariate Normal Distribution	195	(20)
Distribution of the sample mean and covariance matrix	195	(5)
Distributions related to correlation coefficients	200	(4)
Assessing the normality assumption	204	(5)
Transformations to approximate normality	209	(6)
Univariate transformations	209	(2)
Multivariate transformations	211	(1)
Exercises	212	(3)
Inference for the General Linear Model	215	(66)
Properties of least squares estimates	215	(4)
General linear hypotheses	219	(14)
Derivation of and motivation for the F-test	219	(12)
Power of the F-test	231	(1)
Testing independent and orthogonal contrasts	232	(1)
Confidence intervals and multiple comparisons	233	(13)
Joint and marginal confidence intervals	233	(3)
Simultaneous confidence intervals	236	(3)
Multiple comparison procedures	239	(7)
Restricted and reduced models	246	(20)
Nested sequence of hypotheses	246	(17)
Lack of fit test	263	(3)
Non-testable hypotheses	266	(1)
Likelihood based approaches	266	(15)
Maximum likelihood estimation under normality	267	(2)
Elliptically contoured linear model	269	(1)
Model selection criteria	270	(1)
Other types of likelihood analyses	271	(6)
Exercises	277	(4)
Multiple Regression Models	281	(76)
Departures from model assumptions	281	(15)
Graphical procedures	282	(3)
Sequential and partial F-tests	285	(2)
Heteroscedasticity	287	(4)
Serial correlation	291	(4)
Stochastic X matrix	295	(1)
Model selection in regression	296	(8)
Orthogonal and collinear predictors	304	(10)
Orthogonality in regression	304	(3)
Multicollinearity	307	(2)
Ridge regression	309	(4)
Principal components regression	313	(1)
Prediction intervals and calibration	314	(5)
Regression diagnostics	319	(17)
Further properties of the projection matrix	320	(1)
Types of residuals	321	(4)
Outliers and high leverage observations	325	(1)
Diagnostic measures based on influence functions	326	(10)
Dummy variables in regression	336	(3)
Robust regression	339	(5)
Least absolute deviations (LAD) regression	340	(3)
M-regression	343	(1)
Nonparametric regression methods	344	(13)
Additive models	345	(2)
Projection pursuit regression	347	(1)
Neural networks regression	348	(2)
Curve estimation based on wavelet methods	350	(3)
Exercises	353	(4)
Fixed Effects Linear Models	357	(28)
Checking model assumptions	357	(2)
Inference for unbalanced ANOVA models	359	(12)
One-way cell means model	361	(2)
Higher-order overparametrized models	363	(8)
Analysis of covariance	371	(7)
Nonparametric procedures	378	(7)
Kruskal-Wallis procedure	379	(2)
Friedman's procedure	381	(1)
Exercises	381	(4)
Random-Effects and Mixed-Effects Models	385	(22)
One-factor random-effects model	385	(10)
ANOVA method	388	(4)
Maximum likelihood estimation	392	(3)
Restricted maximum likelihood (REML) estimation	395	(1)
Mixed-effects linear models	395	(12)
Extended Gauss-Markov theorem	396	(2)
Estimation procedures	398	(6)
Exercises	404	(3)
Special Topics	407	(26)
Bayesian linear models	407	(4)
Dynamic linear models	411	(5)
Kalman filter equations	412	(3)
Kalman smoothing equations	415	(1)
Longitudinal models	416	(6)
Multivariate models	417	(3)
Two-stage random-effects models	420	(2)
Generalized linear models	422	(11)
Components of GLIM	422	(2)
Estimation approaches	424	(4)
Residuals and model checking	428	(2)
Generalized additive models	430	(1)
Exercises	431	(2)
A Review of Probability Distributions	433	(8)
Solutions to Selected Exercises	441	(8)
References	449	(16)
Author Index	465	(4)
Subject Index	469