A First Course in Linear Model Theory (Texts in Statistical Science)

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A First Course in Linear Model Theory (Texts in Statistical Science)

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  • 製本 Hardcover:ハードカバー版/ページ数 473 p.
  • 言語 ENG,ENG
  • 商品コード 9781584882473
  • DDC分類 519.535

Full Description


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. With an innovative approach, the author's introduces students to the mathematical and statistical concepts and tools that form a foundation for studying the theory and applications of both univariate and multivariate linear models A First Course in Linear Model Theory systematically presents the basic theory behind linear statistical models with motivation from an algebraic as well as a geometric perspective. Through the concepts and tools of matrix and linear algebra and distribution theory, it provides a framework for understanding classical and contemporary linear model theory. It does not merely introduce formulas, but develops in students the art of statistical thinking and inspires learning at an intuitive level by emphasizing conceptual understanding. The authors' fresh approach, methodical presentation, wealth of examples, and introduction to topics beyond the classical theory set this book apart from other texts on linear models. It forms a refreshing and invaluable first step in students' study of advanced linear models, generalized linear models, nonlinear models, and dynamic models.

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 51 (6)
matrices
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 73 (18)
Systems
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 145(19)
properties
Some noncentral distributions 164(8)
Distributions of quadratic forms 172(9)
Alternatives to the multivariate normal 181(14)
distribution
Mixture of normals distribution 181(3)
Spherical distributions 184(1)
Elliptical distributions 185(5)
Exercises 190(5)
Sampling from the Multivariate Normal 195(20)
Distribution
Distribution of the sample mean and 195(5)
covariance matrix
Distributions related to correlation 200(4)
coefficients
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 219(12)
F-test
Power of the F-test 231(1)
Testing independent and orthogonal 232(1)
contrasts
Confidence intervals and multiple 233(13)
comparisons
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 267(2)
normality
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 320(1)
matrix
Types of residuals 321(4)
Outliers and high leverage observations 325(1)
Diagnostic measures based on influence 326(10)
functions
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) 395(1)
estimation
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