A First Course in Linear Model Theory,9781584882473

A First Course in Linear Model Theory

by ;
ISBN13: 9781584882473
ISBN10: 1584882476
Format: Hardcover
Pub. Date: 2001-12-21
Publisher(s): Chapman & Hall/
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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

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