Theory of Statistical Inference

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 d-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.

Preface 1 Distribution Theory 1.1 Introduction 1.2 Probability Measures 1.3 Some Important Theorems of Probability 1.4 Commonly Used Distributions 1.5 Stochastic Order Relations 1.6 Quantiles 1.7 Inversion of the CDF 1.8 Transformations of Random Variables 1.9 Moment Generating Functions 1.10 Moments and Cumulants 1.11 Problems 2 Multivariate Distributions 2.1 Introduction 2.2 Parametric Classes of Multivariate Distributions 2.3 Multivariate Transformations 2.4 Order Statistics 2.5 Quadratic Forms, Idempotent Matrices and Cochran’s Theorem 2.6 MGF and CGF of Independent Sums 2.7 Multivariate Extensions of the MGF 2.8 Problems 3 Statistical Models 3.1 Introduction 3.2 Parametric Families for Statistical Inference 3.3 Location-Scale Parameter Models 3.4 Regular Families 3.5 Fisher Information 3.6 Exponential Families 3.7 Sufficiency 3.8 Complete and Ancillary Statistics 3.9 Conditional Models and Contingency Tables 3.10 Bayesian Models 3.11 Indifference, Invariance and Bayesian Prior Distributions 3.12 Nuisance Parameters 3.13 Principles of Inference 3.14 Problems 4 Methods of Estimation 4.1 Introduction 4.2 Unbiased Estimators 4.3 Method of Moments Estimators 4.4 Sample Quantiles and Percentiles 4.5 Maximum Likelihood Estimation 4.6 Confidence Sets 4.7 Equivariant Versus Shrinkage Estimation 4.8 Bayesian Estimation 4.9 Problems 5 Hypothesis Testing 5.1 Introduction 5.2 Basic Definitions 5.3 Principles of Hypothesis Tests 5.4 The Observed Level of Significance (P-Values) 5.5 One and Two Sided Tests 5.6 Hypothesis Tests and Pivots 5.7 Likelihood Ratio Tests 5.8 Similar Tests 5.9 Problems 6 Linear Models 6.1 Introduction 6.2 Linear Models - Definition 6.3 Best Linear Unbiased Estimators (BLUE) 6.4 Least-squares Estimators, BLUEs and Projection Matrices 6.5 Ordinary and Generalized Least-Squares Estimators 6.6 ANOVA Decomposition and the F Test for Linear Models 6.7 The F Test for One-Way ANOVA 6.8 Simultaneous Confidence Intervals 6.9 Multiple Linear Regression 6.10 Problems 7 Decision Theory 7.1 Introduction 7.2 Ranking Estimators by MSE 7.3 Prediction 7.4 The Structure of Decision Theoretic Inference 7.5 Loss and Risk 7.6 Uniformly Minimum Risk Estimators (The Location-Scale Model 7.7 Some Principles of Admissibility 7.8 Admissibility for Exponential Families (Karlin’s Theorem) 7.9 Bayes Decision Rules 7.10 Admissibility and Optimality 7.11 Problems 8 Uniformly Minimum Variance Unbiased (UMVU) Estimation 8.1 Introduction 8.2 Definition of UMVUE’s 8.3 UMVUE’s and Sufficiency 8.4 Methods of Deriving UMVUEs 8.5 Nonparametric Estimation and U-statistics 8.6 Rank Based Measures of Correlation 8.7 Problems 9 Group Structure and Invariant Inference 9.1 Introduction 9.2 MRE Estimators for Location Parameters 9.3 MRE Estimators for Scale Parameters 9.4 Invariant Density Families 9.5 Some Applications of Invariance 9.6 Invariant Hypothesis Tests 9.7 Problems 10 The Neyman-Pearson Lemma 10.1 Introduction 10.2 Hypothesis Test as Decision Rules 10.3 Neyman-Pearson (NP) Tests 10.4 Monotone Likelihood Ratios (MLR) 10.5 The Generalized Neyman-Pearson Lemma 10.6 Invariant Hypothesis Tests 10.7 Permutation Invariant Tests 10.8 Problems 11 Limit Theorems 11.1 Introduction 11.2 Limits of Sequences of Random Variables 11.3 Limits of Expected Values 11.4 Uniform Integrability 11.5 The Law of Large Numbers 11.6 Weak Convergence 11.7 Multivariate Extensions of Limit Theorems 11.8 The Continuous Mapping Theorem 11.9 MGFs, CGFs and Weak Convergence 11.10 The Central Limit Theorem for Triangular Arrays 11.11 Weak Convergence of Random Vectors 11.12 Problems 12 Large Sample Estimation - Basic Principles 12.1 Introduction 12.2 The _-Method 12.3 Variance Stabilizing Transformations 12.4 The _-Method and Higher Order Approximations 12.5 The Multivariate _-Method 12.6 Approximating the Distributions of Sample Quantiles: The Bahadur Representation Theorem 12.7 A Central Limit Theorem for U-statistics 12.8 The Information Inequality 12.9 Asymptotic Efficiency 12.10 Problems 13 Asymptotic Theory for Estimating Equations 13.1 Introduction 13.2 Consistency and Asymptotic Normality of M-estimators 13.3 Asymptotic Theory of MLEs 13.4 A General Form for Regression Models 13.5 Nonlinear Regression 13.6 Generalized Linear Models (GLMs) 13.7 Generalized Estimating Equations (GEE) 13.8 Consistency of M-estimators 13.9 Asymptotic Distribution of ˆ_n 13.10 Regularity Conditions for Estimating Equations 13.11 Problems 14 Large Sample Hypothesis Testing 14.1 Introduction 14.2 Model Assumptions 14.3 Large Sample Tests for Simple Hypotheses 14.4 Nuisance Parameters and Composite Null Hypotheses 14.5 A Comparison of the LR, Wald and Score Tests 14.6 Pearson’s _2 Test for Independence in Contingency Tables 14.7 Estimating Power for Approximate _2 Tests 14.8 Problems A Parametric Classes of Densities B Topics in Linear Algebra B.1 Numbers B.2 Equivalence Relations B.3 Vector Spaces B.4 Matrices B.5 Dimension of a Subset of Rd C Topics in Real Analysis and Measure Theory C.1 Metric spaces C.2 Measure Theory C.3 Integration C.4 Exchange of Integration and Differentiation C.5 The Gamma and Beta Functions C.6 Stirling’s Approximation of the Factorial C.7 The Gradient Vector and the Hessian Matrix C.8 Normed Vector Spaces C.9 Taylor’s Theorem D Group Theory D.1 Definition of a Group D.2 Subgroups D.3 Group Homomorphisms D.4 Transformation Groups D.5 Orbits and Maximal Invariants Bibliography Index

Anthony Almudevar is an Associate Professor of Biostatistics and Computational Biology at the University of Rochester. His research interests include statistical methodology, graphical models, bioinformatics, optimization and control theory. Other published volumes include Almudevar A (2014) Approximate Iterative Algorithms, CRC Press, and Statistical Modeling for Biological Systems: In Memory of Andrei Yakovlev, Anthony Almudevar, David Oakes and Jack Hall, editors (2020), Springer.