Table Of ContentCover Page Page: i
Half-Title Page Page: i
Series Page Page: ii
Title Page Page: iii
Copyright Page Page: iv
Contents Page: v
Preface Page: xi
1 Distribution Theory Page: 1
1.1 Introduction Page: 1
1.2 Probability Measures Page: 1
1.3 Some Important Theorems of Probability Page: 7
1.4 Commonly Used Distributions Page: 10
1.5 Stochastic Order Relations Page: 16
1.6 Quantiles Page: 17
1.7 Inversion of the CDF Page: 19
1.8 Transformations of Random Variables Page: 21
1.9 Moment Generating Functions Page: 23
1.10 Moments and Cumulants Page: 27
1.11 Problems Page: 30
2 Multivariate Distributions Page: 37
2.1 Introduction Page: 37
2.2 Parametric Classes of Multivariate Distributions Page: 37
2.3 Multivariate Transformations Page: 40
2.4 Order Statistics Page: 42
2.5 Quadratic Forms, Idempotent Matrices and Cochran's Theorem Page: 44
2.6 MGF and CGF of Independent Sums Page: 49
2.7 Multivariate Extensions of the MGF Page: 51
2.8 Problems Page: 51
3 Statistical Models Page: 57
3.1 Introduction Page: 57
3.2 Parametric Families for Statistical Inference Page: 58
3.3 Location-Scale Parameter Models Page: 61
3.4 Regular Families Page: 69
3.5 Fisher Information Page: 69
3.6 Exponential Families Page: 72
3.7 Sufficiency Page: 78
3.8 Complete and Ancillary Statistics Page: 82
3.9 Conditional Models and Contingency Tables Page: 88
3.10 Bayesian Models Page: 89
3.11 Indifference, Invariance and Bayesian Prior Distributions Page: 91
3.12 Nuisance Parameters Page: 95
3.13 Principles of Inference Page: 95
3.14 Problems Page: 98
4 Methods of Estimation Page: 105
4.1 Introduction Page: 105
4.2 Unbiased Estimators Page: 106
4.3 Method of Moments Estimators Page: 107
4.4 Sample Quantiles and Percentiles Page: 108
4.5 Maximum Likelihood Estimation Page: 109
4.6 Confidence Sets Page: 116
4.7 Equivariant Versus Shrinkage Estimation Page: 122
4.8 Bayesian Estimation Page: 123
4.9 Problems Page: 127
5 Hypothesis Testing Page: 133
5.1 Introduction Page: 133
5.2 Basic Definitions Page: 134
5.3 Principles of Hypothesis Tests Page: 135
5.4 The Observed Level of Significance (P-Values) Page: 137
5.5 One- and Two-Sided Tests Page: 138
5.6 Unbiasedness and Stochastic Ordering Page: 139
5.7 Hypothesis Tests and Pivots Page: 140
5.8 Likelihood Ratio Tests Page: 141
5.9 Similar Tests Page: 146
5.10 Problems Page: 147
6 Linear Models Page: 155
6.1 Introduction Page: 155
6.2 Linear Models – Definition Page: 155
6.3 Best Linear Unbiased Estimators (BLUE) Page: 158
6.4 Least Squares Estimators, BLUEs and Projection Matrices Page: 161
6.5 Ordinary and Generalized Least Squares Estimators Page: 163
6.6 ANOVA Decomposition and the F Test for Linear Models Page: 168
6.7 One- and Two-Way ANOVA Page: 174
6.8 Multiple Linear Regression Page: 181
6.9 Constrained Least Squares Estimation Page: 187
6.10 Simultaneous Confidence Intervals Page: 190
6.11 Problems Page: 196
7 Decision Theory Page: 207
7.1 Introduction Page: 207
7.2 Ranking Estimators by MSE Page: 208
7.3 Prediction Page: 211
7.4 The Structure of Decision Theoretic Inference Page: 215
7.5 Loss and Risk Page: 218
7.6 Uniformly Minimum Risk Estimators (The Location-Scale Model) Page: 221
7.7 Some Principles of Admissibility Page: 224
7.8 Admissibility for Exponential Families (Karlin's Theorem) Page: 226
7.9 Bayes Decision Rules Page: 228
7.10 Admissibility and Optimality Page: 232
7.11 Problems Page: 235
8 Uniformly Minimum Variance Unbiased (UMVU) Estimation Page: 241
8.1 Introduction Page: 241
8.2 Definition of UMVUE's Page: 241
8.3 UMVUE's and Sufficiency Page: 243
8.4 Methods of Deriving UMVUEs Page: 245
8.5 Nonparametric Estimation and U-statistics Page: 247
8.6 Rank Based Measures of Correlation Page: 252
8.7 Problems Page: 254
9 Group Structure and Invariant Inference Page: 257
9.1 Introduction Page: 257
9.2 MRE Estimators for Location Parameters Page: 258
9.3 MRE Estimators for Scale Parameters Page: 264
9.4 Invariant Density Families Page: 270
9.5 Some Applications of Invariance Page: 274
9.6 Invariant Hypothesis Tests Page: 278
9.7 Problems Page: 283
10 The Neyman-Pearson Lemma Page: 289
10.1 Introduction Page: 289
10.2 Hypothesis Tests as Decision Rules Page: 289
10.3 Neyman-Pearson (NP) Tests Page: 290
10.4 Monotone Likelihood Ratios (MLR) Page: 294
10.5 The Generalized Neyman-Pearson Lemma Page: 295
10.6 Invariant Hypothesis Tests Page: 301
10.7 Permutation Invariant Tests Page: 303
10.8 Problems Page: 310
11 Limit Theorems Page: 315
11.1 Introduction Page: 315
11.2 Limits of Sequences of Random Variables Page: 315
11.3 Limits of Expected Values Page: 318
11.4 Uniform Integrability Page: 319
11.5 The Law of Large Numbers Page: 321
11.6 Weak Convergence Page: 324
11.7 Multivariate Extensions of Limit Theorems Page: 326
11.8 The Continuous Mapping Theorem Page: 329
11.9 MGFs, CGFs and Weak Convergence Page: 330
11.10 The Central Limit Theorem for Triangular Arrays Page: 332
11.11 Weak Convergence of Random Vectors Page: 334
11.12 Problems Page: 335
12 Large Sample Estimation –- Basic Principles Page: 341
12.1 Introduction Page: 341
12.2 The δ-Method Page: 341
12.3 Variance Stabilizing Transformations Page: 344
12.4 The δ-Method and Higher-Order Approximations Page: 347
12.5 The Multivariate δ-Method Page: 353
12.6 Approximating the Distributions of Sample Quantiles: The Bahadur Representation Theorem Page: 354
12.7 A Central Limit Theorem for U-statistics Page: 357
12.8 The Information Inequality Page: 358
12.9 Asymptotic Efficiency Page: 362
12.10 Problems Page: 364
13 Asymptotic Theory for Estimating Equations Page: 371
13.1 Introduction Page: 371
13.2 Consistency and Asymptotic Normality of M-Estimators Page: 372
13.3 Asymptotic Theory of MLEs Page: 375
13.4 A General Form for Regression Models Page: 376
13.5 Nonlinear Regression Page: 378
13.6 Generalized Linear Models (GLM) Page: 379
13.7 Generalized Estimating Equations (GEE) Page: 385
13.8 Existence and Consistency of M-Estimators Page: 387
13.9 Asymptotic Distribution of θ^n Page: 389
13.10 Regularity Conditions for Estimating Equations Page: 390
13.11 Problems Page: 391
14 Large Sample Hypothesis Testing Page: 395
14.1 Introduction Page: 395
14.2 Model Assumptions Page: 395
14.3 Large Sample Tests for Simple Null Hypotheses Page: 397
14.4 Nuisance Parameters and Composite Null Hypotheses Page: 402
14.5 Pearson's χ2 Test for Independence in Contingency Tables Page: 407
14.6 A Comparison of the LR, Wald and Score Tests Page: 409
14.7 Confidence Sets Page: 410
14.8 Estimating Power for Approximate χ2 Tests Page: 411
14.9 Problems Page: 411
A Parametric Classes of Densities Page: 415
B Topics in Linear Algebra Page: 417
B.1 Numbers Page: 417
B.2 Equivalence Relations Page: 418
B.3 Vector Spaces Page: 418
B.4 Matrices Page: 419
B.5 Dimension of a Subset of ℝd Page: 425
C Topics in Real Analysis and Measure Theory Page: 427
C.1 Metric Spaces Page: 427
C.2 Measure Theory Page: 428
C.3 Integration Page: 429
C.4 Exchange of Integration and Differentiation Page: 430
C.5 The Gamma and Beta Functions Page: 431
C.6 Stirling's Approximation of the Factorial Page: 432
C.7 The Gradient Vector and the Hessian Matrix Page: 432
C.8 Normed Vector Spaces Page: 433
C.9 Taylor's Remainder Theorem Page: 435
D Group Theory Page: 437
D.1 Definition of a Group Page: 437
D.2 Subgroups Page: 438
D.3 Group Homomorphisms Page: 439
D.4 Transformation Groups Page: 440
D.5 Orbits and Maximal Invariants Page: 442
Bibliography Page: 445
Index Page: 453
Description:Theory of Statistical Inference is designed as a reference on statistical inference for researchers and students at the graduate or advanced undergraduate level. It presents a unified treatment of the foundational ideas of modern statistical inference, and would be suitable for a core course in a graduate program in statistics or biostatistics. The emphasis is on the application of mathematical theory to the problem of inference, leading to an optimization theory allowing the choice of those statistical methods yielding the most efficient use of data. The book shows how a small number of key concepts, such as sufficiency, invariance, stochastic ordering, decision theory and vector space algebra play a recurring and unifying role. The volume can be divided into four sections. Part I provides a review of the required distribution theory. Part II introduces the problem of statistical inference. This includes the definitions of the exponential family, invariant and Bayesian models. Basic concepts of estimation, confidence intervals and hypothesis testing are introduced here. Part III constitutes the core of the volume, presenting a formal theory of statistical inference. Beginning with decision theory, this section then covers uniformly minimum variance unbiased (UMVU) estimation, minimum risk equivariant (MRE) estimation and the Neyman-Pearson test. Finally, Part IV introduces large sample theory. This section begins with stochastic limit theorems, the δ-method, the Bahadur representation theorem for sample quantiles, large sample U-estimation, the Cramér-Rao lower bound and asymptotic efficiency. A separate chapter is then devoted to estimating equation methods. The volume ends with a detailed development of large sample hypothesis testing, based on the likelihood ratio test (LRT), Rao score test and the Wald test. Features This volume includes treatment of linear and nonlinear regression models, ANOVA models, generalized linear models (GLM) and generalized estimating equations (GEE). An introduction to decision theory (including risk, admissibility, classification, Bayes and minimax decision rules) is presented. The importance of this sometimes overlooked topic to statistical methodology is emphasized. The volume emphasizes throughout the important role that can be played by group theory and invariance in statistical inference. Nonparametric (rank-based) methods are derived by the same principles used for parametric models and are therefore presented as solutions to well-defined mathematical problems, rather than as robust heuristic alternatives to parametric methods. Each chapter ends with a set of theoretical and applied exercises integrated with the main text. Problems involving R programming are included. Appendices summarize the necessary background in analysis, matrix algebra and group theory.
