[下载]wiley08《统计学中的线性模型》（Linear Models in Statistics）

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收藏 2008-01-19

Linear Models in Statistics (Wiley Series in Probability and Statistics)
by Alvin C. Rencher (Author), G. Bruce Schaalje (Author)

Hardcover: 672 pages

Publisher: Wiley-Interscience; 2 edition (January 2, 2008)

Language: English

Review
"Rencher...offers a textbook for a one-semester advanced undergraduate or beginning graduate course.... He includes more material than can actually squeeze into one semester...a good idea in statistics." (SciTech Book News, Vol. 24, No. 4, December 2000)

"An excellent book. Highly recommended. Upper-division undergraduate and graduate students; professionals." (Choice, Vol. 38, No. 7, March 2001)

"I would recommend the book to anyone as a reference book for the topics covered.... The book should also be a strong candidate for any M.S. course in linear models because of the numerous exercises with solutions and clear writing style." (Technometrics, Vol. 42, No. 4, May 2001)

"Rencher's textbook is certainly of interest for students and instructors looking for a mathematical introduction to linear statistical models." (Statistics & Decisions, Volume 19, No 2, 2001)

"...courses that go by the name "linear models" cover a combination of linear model theory, regression diagnostic, analysis of variance and more complex models that use linear models as a stepping stone. This book is appropriate for such courses...the collection of exercises adds to the book's value as a textbook." (Journal of the American Statistical Association, September 2001)

"Gives a solid theoretical foundation to standard topics..." (American Mathematical Monthly, November 2001) --This text refers to the Hardcover edition.

Choice, Vol. 38, No. 7, March 2001
"An excellent book. Highly recommended. Upper-division undergraduate and graduate students; professionals." --This text refers to the Hardcover edition.

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Linear Models in Statistics, 2nd Ed.pdf

CONTENTS
Preface xiii
1 Introduction 1
1.1 Simple Linear Regression Model 1
1.2 Multiple Linear Regression Model 2
1.3 Analysis-of-Variance Models 3
2 Matrix Algebra 5
2.1 Matrix and Vector Notation 5
2.1.1 Matrices, Vectors, and Scalars 5
2.1.2 Matrix Equality 6
2.1.3 Transpose 7
2.1.4 Matrices of Special Form 7
2.2 Operations 9
2.2.1 Sum of Two Matrices or Two Vectors 9
2.2.2 Product of a Scalar and a Matrix 10
2.2.3 Product of Two Matrices or Two Vectors 10
2.2.4 Hadamard Product of Two
Matrices or Two Vectors 16
2.3 Partitioned Matrices 16
2.4 Rank 19
2.5 Inverse 21
2.6 Positive Definite Matrices 24
2.7 Systems of Equations 28
2.8 Generalized Inverse 32
2.8.1 Definition and Properties 33
2.8.2 Generalized Inverses and Systems of Equations 36
2.9 Determinants 37
2.10 Orthogonal Vectors and Matrices 41
2.11 Trace 44
2.12 Eigenvalues and Eigenvectors 46
2.12.1 Definition 46
2.12.2 Functions of a Matrix 49
2.12.3 Products 50
2.12.4 Symmetric Matrices 51
2.12.5 Positive Definite and Semidefinite Matrices 53
2.13 Idempotent Matrices 54
2.14 Vector and Matrix Calculus 56
2.14.1 Derivatives of Functions of Vectors and Matrices 56
2.14.2 Derivatives Involving Inverse Matrices and Determinants 58
2.14.3 Maximization or Minimization of a Function of a Vector 60
3 Random Vectors and Matrices 69
3.1 Introduction 69
3.2 Means, Variances, Covariances, and Correlations 70
3.3 Mean Vectors and Covariance Matrices for Random Vectors 75
3.3.1 Mean Vectors 75
3.3.2 Covariance Matrix 75
3.3.3 Generalized Variance 77
3.3.4 Standardized Distance 77
3.4 Correlation Matrices 77
3.5 Mean Vectors and Covariance Matrices for
Partitioned Random Vectors 78
3.6 Linear Functions of Random Vectors 79
3.6.1 Means 80
3.6.2 Variances and Covariances 81
4 Multivariate Normal Distribution 87
4.1 Univariate Normal Density Function 87
4.2 Multivariate Normal Density Function 88
4.3 Moment Generating Functions 90
4.4 Properties of the Multivariate Normal Distribution 92
4.5 Partial Correlation 100
5 Distribution of Quadratic Forms in y 105
5.1 Sums of Squares 105
5.2 Mean and Variance of Quadratic Forms 107
5.3 Noncentral Chi-Square Distribution 112
5.4 Noncentral F and t Distributions 114
5.4.1 Noncentral F Distribution 114
5.4.2 Noncentral t Distribution 116
5.5 Distribution of Quadratic Forms 117
5.6 Independence of Linear Forms and Quadratic Forms 119
6 Simple Linear Regression 127
6.1 The Model 127
6.2 Estimation of b0, b1, and s2 128
6.3 Hypothesis Test and Confidence Interval for b1 132
6.4 Coefficient of Determination 133
7 Multiple Regression: Estimation 137
7.1 Introduction 137
7.2 The Model 137
7.3 Estimation of b and s2 141
7.3.1 Least-Squares Estimator for b 145
7.3.2 Properties of the Least-Squares Estimator b ˆ 141
7.3.3 An Estimator for s2 149
7.4 Geometry of Least-Squares 151
7.4.1 Parameter Space, Data Space, and Prediction Space 152
7.4.2 Geometric Interpretation of the Multiple
Linear Regression Model 153
7.5 The Model in Centered Form 154
7.6 Normal Model 157
7.6.1 Assumptions 157
7.6.2 Maximum Likelihood Estimators for b and s2 158
7.6.3 Properties of b ˆ and sˆ 2 159
7.7 R2 in Fixed-x Regression 161
7.8 Generalized Least-Squares: cov(y) ¼ s2V 164
7.8.1 Estimation of b and s2 when cov(y) ¼ s2V 164
7.8.2 Misspecification of the Error Structure 167
7.9 Model Misspecification 169
7.10 Orthogonalization 174
8 Multiple Regression: Tests of Hypotheses
and Confidence Intervals 185
8.1 Test of Overall Regression 185
8.2 Test on a Subset of the b Values 189
8.3 F Test in Terms of R2 196
8.4 The General Linear Hypothesis Tests for H0:
Cb ¼ 0 and H0: Cb ¼ t 198
8.4.1 The Test for H0: Cb ¼ 0 198
8.4.2 The Test for H0: Cb ¼ t 203
8.5 Tests on bj and a0b 204
8.5.1 Testing One bj or One a0b 204
8.5.2 Testing Several bj or a0ib Values 205
CONTENTS vii
8.6 Confidence Intervals and Prediction Intervals 209
8.6.1 Confidence Region for b 209
8.6.2 Confidence Interval for bj 210
8.6.3 Confidence Interval for a0b 211
8.6.4 Confidence Interval for E(y) 211
8.6.5 Prediction Interval for a Future Observation 213
8.6.6 Confidence Interval for s2 215
8.6.7 Simultaneous Intervals 215
8.7 Likelihood Ratio Tests 217
9 Multiple Regression: Model Validation and Diagnostics 227
9.1 Residuals 227
9.2 The Hat Matrix 230
9.3 Outliers 232
9.4 Influential Observations and Leverage 235
10 Multiple Regression: Random x’s 243
10.1 Multivariate Normal Regression Model 244
10.2 Estimation and Testing in Multivariate Normal Regression 245
10.3 Standardized Regression Coefficents 249
10.4 R2 in Multivariate Normal Regression 254
10.5 Tests and Confidence Intervals for R2 258
10.6 Effect of Each Variable on R2 262
10.7 Prediction for Multivariate Normal or Nonnormal Data 265
10.8 Sample Partial Correlations 266
11 Multiple Regression: Bayesian Inference 277
11.1 Elements of Bayesian Statistical Inference 277
11.2 A Bayesian Multiple Linear Regression Model 279
11.2.1 A Bayesian Multiple Regression Model
with a Conjugate Prior 280
11.2.2 Marginal Posterior Density of b 282
11.2.3 Marginal Posterior Densities of t and s2 284
11.3 Inference in Bayesian Multiple Linear Regression 285
11.3.1 Bayesian Point and Interval Estimates of
Regression Coefficients 285
11.3.2 Hypothesis Tests for Regression Coefficients
in Bayesian Inference 286
11.3.3 Special Cases of Inference in Bayesian Multiple
Regression Models 286
11.3.4 Bayesian Point and Interval Estimation of s2 287
11.4 Bayesian Inference through Markov Chain
Monte Carlo Simulation 288
11.5 Posterior Predictive Inference 290
12 Analysis-of-Variance Models 295
12.1 Non-Full-Rank Models 295
12.1.1 One-Way Model 295
12.1.2 Two-Way Model 299
12.2 Estimation 301
12.2.1 Estimation of b 302
12.2.2 Estimable Functions of b 305
12.3 Estimators 309
12.3.1 Estimators of l0b 309
12.3.2 Estimation of s2 313
12.3.3 Normal Model 314
12.4 Geometry of Least-Squares in the
Overparameterized Model 316
12.5 Reparameterization 318
12.6 Side Conditions 320
12.7 Testing Hypotheses 323
12.7.1 Testable Hypotheses 323
12.7.2 Full-Reduced-Model Approach 324
12.7.3 General Linear Hypothesis 326
12.8 An Illustration of Estimation and Testing 329
12.8.1 Estimable Functions 330
12.8.2 Testing a Hypothesis 331
12.8.3 Orthogonality of Columns of X 333
13 One-Way Analysis-of-Variance: Balanced Case 339
13.1 The One-Way Model 339
13.2 Estimable Functions 340
13.3 Estimation of Parameters 341
13.3.1 Solving the Normal Equations 341
13.3.2 An Estimator for s2 343
13.4 Testing the Hypothesis H0: m1 ¼ m2 ¼ . . . ¼ mk 344
13.4.1 Full–Reduced-Model Approach 344
13.4.2 General Linear Hypothesis 348
13.5 Expected Mean Squares 351
13.5.1 Full-Reduced-Model Approach 352
13.5.2 General Linear Hypothesis 354
13.6 Contrasts 357
13.6.1 Hypothesis Test for a Contrast 357
13.6.2 Orthogonal Contrasts 358
13.6.3 Orthogonal Polynomial Contrasts 363
14 Two-Way Analysis-of-Variance: Balanced Case 377
14.1 The Two-Way Model 377
14.2 Estimable Functions 378
14.3 Estimators of l0b and s2 382
14.3.1 Solving the Normal Equations and Estimating l0b 382
14.3.2 An Estimator for s2 384
14.4 Testing Hypotheses 385
14.4.1 Test for Interaction 385
14.4.2 Tests for Main Effects 395
14.5 Expected Mean Squares 403
14.5.1 Sums-of-Squares Approach 403
14.5.2 Quadratic Form Approach 405
15 Analysis-of-Variance: The Cell Means Model for
Unbalanced Data 413
15.1 Introduction 413
15.2 One-Way Model 415
15.2.1 Estimation and Testing 415
15.2.2 Contrasts 417
15.3 Two-Way Model 421
15.3.1 Unconstrained Model 421
15.3.2 Constrained Model 428
15.4 Two-Way Model with Empty Cells 432
16 Analysis-of-Covariance 443
16.1 Introduction 443
16.2 Estimation and Testing 444
16.2.1 The Analysis-of-Covariance Model 444
16.2.2 Estimation 446
16.2.3 Testing Hypotheses 448
16.3 One-Way Model with One Covariate 449
16.3.1 The Model 449
16.3.2 Estimation 449
16.3.3 Testing Hypotheses 450
x CONTENTS
16.4 Two-Way Model with One Covariate 457
16.4.1 Tests for Main Effects and Interactions 458
16.4.2 Test for Slope 462
16.4.3 Test for Homogeneity of Slopes 463
16.5 One-Way Model with Multiple Covariates 464
16.5.1 The Model 464
16.5.2 Estimation 465
16.5.3 Testing Hypotheses 468
16.6 Analysis-of-Covariance with Unbalanced Models 473
17 Linear Mixed Models 479
17.1 Introduction 479
17.2 The Linear Mixed Model 479
17.3 Examples 481
17.4 Estimation of Variance Components 486
17.5 Inference for b 490
17.5.1 An Estimator for b 490
17.5.2 Large-Sample Inference for Estimable Functions of b 491
17.5.3 Small-Sample Inference for Estimable Functions of b 491
17.6 Inference for the ai Terms 497
17.7 Residual Diagnostics 501
18 Additional Models 507
18.1 Nonlinear Regression 507
18.2 Logistic Regression 508
18.3 Loglinear Models 511
18.4 Poisson Regression 512
18.5 Generalized Linear Models 513
Appendix A Answers and Hints to the Problems 517
References 653
Index 663

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