Table Of Content

Cover 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.