Testing Statistical Hypotheses

Preface to the Fourth Edition
Contents
Volume I Finite-Sample Theory
1 The General Decision Problem
1.1 Statistical Inference and Statistical Decisions
1.2 Specification of a Decision Problem
1.3 Randomization; Choice of Experiment
1.4 Optimum Procedures
1.5 Invariance and Unbiasedness
1.6 Bayes and Minimax Procedures
1.7 Maximum Likelihood
1.8 Complete Classes
1.9 Sufficient Statistics
1.10 Problems
1.11 Notes
2 The Probability Background
2.1 Probability and Measure
2.2 Integration
2.3 Statistics and Subfields
2.4 Conditional Expectation and Probability
2.5 Conditional Probability Distributions
2.6 Characterization of Sufficiency
2.7 Exponential Families
2.8 Problems
2.9 Notes
3 Uniformly Most Powerful Tests
3.1 Stating the Problem
3.2 The Neyman–Pearson Fundamental Lemma
3.3 p-values
3.4 Distributions with Monotone Likelihood Ratio
3.5 Confidence Bounds
3.6 A Generalization of the Fundamental Lemma
3.7 Two-Sided Hypotheses
3.8 Least Favorable Distributions
3.9 Applications to Normal Distributions
3.9.1 Univariate Normal Models
3.9.2 Multivariate Normal Models
3.10 Problems
3.11 Notes
4 Unbiasedness: Theory and First Applications
4.1 Unbiasedness for Hypothesis Testing
4.2 One-Parameter Exponential Families
4.3 Similarity and Completeness
4.4 UMP Unbiased Tests for Multiparameter Exponential Families
4.5 Comparing Two Poisson or Binomial Populations
4.6 Testing for Independence in a 2times2 Table
4.7 Alternative Models for 2times2 Tables
4.8 Some Three-Factor Contingency Tables
4.9 The Sign Test
4.10 Problems
4.11 Notes
5 Unbiasedness: Applications to Normal Distributions; Confidence Intervals
5.1 Statistics Independent of a Sufficient Statistic
5.2 Testing the Parameters of a Normal Distribution
5.3 Comparing the Means and Variances of Two Normal Distributions
5.4 Confidence Intervals and Families of Tests
5.5 Unbiased Confidence Sets
5.6 Regression
5.7 Bayesian Confidence Sets
5.8 Permutation Tests
5.9 Most Powerful Permutation Tests
5.10 Randomization as a Basis For Inference
5.11 Permutation Tests and Randomization
5.12 Randomization Model and Confidence Intervals
5.13 Testing for Independence in a Bivariate Normal Distribution
5.14 Problems
5.15 Notes
6 Invariance
6.1 Symmetry and Invariance
6.2 Maximal Invariants
6.3 Uniformly Most Powerful Invariant Tests
6.4 Sample Inspection by Variables
6.5 Almost Invariance
6.6 Unbiasedness and Invariance
6.7 Admissibility
6.8 Rank Tests
6.9 The Two-Sample Problem
6.10 The Hypothesis of Symmetry
6.11 Equivariant Confidence Sets
6.12 Average Smallest Equivariant Confidence Sets
6.13 Confidence Bands for a Distribution Function
6.14 Problems
6.15 Notes
7 Linear Hypotheses
7.1 A Canonical Form
7.2 Linear Hypotheses and Least Squares
7.3 Tests of Homogeneity
7.4 Two-Way Layout: One Observation Per Cell
7.5 Two-Way Layout: m Observations Per Cell
7.6 Regression
7.7 Random-Effects Model: One-Way Classification
7.8 Nested Classifications
7.9 Multivariate Extensions
7.10 Problems
7.11 Notes
8 The Minimax Principle
8.1 Tests with Guaranteed Power
8.2 Further Examples
8.3 Comparing Two Approximate Hypotheses
8.4 Maximin Tests and Invariance
8.5 The Hunt–Stein Theorem
8.6 Most Stringent Tests
8.7 Monotone Tests
8.8 Problems
8.9 Notes
9 Multiple Testing and Simultaneous Inference
9.1 Introduction and the FWER
9.1.1 Basic Framework
9.1.2 Single-Step Methods
9.1.3 Stepwise Methods and the Holm Method
9.2 The Closure Method
9.2.1 The Basic Method and Some Examples
9.2.2 Simes' Identity and Hommel's Method
9.2.3 The Higher Criticism and Other Joint Tests
9.2.4 Coherence and Cosonance
9.3 False Discovery Rate and Other Generalized Error Rates
9.3.1 Introduction to Various Error Rates
9.3.2 False Discovery Rate
9.3.3 Control of the k-FWER
9.3.4 Control of the False Discovery Proportion
9.4 Maximin Procedures
9.5 The Hypothesis of Homogeneity
9.6 Scheffé's S-Method: A Special Case
9.7 Scheffé's S-Method for General Linear Models
9.8 Problems
9.9 Notes
10 Conditional Inference
10.1 Mixtures of Experiments
10.2 Ancillary Statistics
10.3 Optimal Conditional Tests
10.4 Relevant Subsets
10.5 Problems
10.6 Notes
Volume II Asymptotic Theory
11 Basic Large-Sample Theory
11.1 Introduction
11.2 Weak Convergence and Central Limit Theorems
11.3 Convergence in Probability and Applications
11.4 Almost Sure Convergence
11.5 Problems
11.6 Notes
12 Extensions of the CLT to Sums of Dependent Random Variables
12.1 Introduction
12.2 Random Sampling Without Replacement from a Finite Population
12.3 U-Statistics
12.4 Stationary Mixing Processes
12.5 Stein's Method
12.6 Problems
12.7 Notes
13 Applications to Inference
13.1 Introduction
13.2 Robustness of Some Classical Tests
13.2.1 Effect of Distribution
13.2.2 Effect of Dependence
13.2.3 Robustness in Linear Models
13.3 Edgeworth Expansions
13.4 Nonparametric Inference for the Mean
13.4.1 Uniform Behavior of t-test
13.4.2 A Result of Bahadur and Savage
13.4.3 Alternative Tests
13.5 Testing Many Means: The Gaussian Sequence Model
13.5.1 Chi-Squared Test
13.5.2 Maximin Test for Sparse Alternatives
13.5.3 Test Based on Maximum and Bonferroni
13.5.4 Some Comparisons and the Higher Criticism
13.6 Problems
13.7 Notes
14 Quadratic Mean Differentiable Families
14.1 Introduction
14.2 Quadratic Mean Differentiability (q.m.d.)
14.3 Contiguity
14.4 Likelihood Methods in Parametric Models
14.4.1 Efficient Likelihood Estimation
14.4.2 Wald Tests and Confidence Regions
14.4.3 Rao Score Tests
14.4.4 Likelihood Ratio Tests
14.5 Problems
14.6 Notes
15 Large-Sample Optimality
15.1 Testing Sequences, Metrics, and Inequalities
15.2 Asymptotic Relative Efficiency
15.3 AUMP Tests in Univariate Models
15.4 Asymptotically Normal Experiments
15.5 Applications to Parametric Models
15.5.1 One-Sided Hypotheses
15.5.2 Equivalence Hypotheses
15.5.3 Multisided Hypotheses
15.6 Applications to Nonparametric Models
15.6.1 Nonparametric Mean
15.6.2 Nonparametric Testing of Functionals
15.7 Problems
15.8 Notes
16 Testing Goodness of Fit
16.1 Introduction
16.2 The Kolmogorov–Smirnov Test
16.2.1 Simple Null Hypothesis
16.2.2 Extensions of the Kolmogorov–Smirnov Test
16.3 Pearson's Chi-Squared Statistic
16.3.1 Simple Null Hypothesis
16.3.2 Chi-Squared Test of Uniformity
16.3.3 Composite Null Hypothesis
16.4 Neyman's Smooth Tests
16.4.1 Fixed k Asymptotics
16.4.2 Neyman's Smooth Tests With Large k
16.5 Weighted Quadratic Test Statistics
16.6 Global Behavior of Power Functions
16.7 Problems
16.8 Notes
17 Permutation and Randomization Tests
17.1 Introduction
17.2 Permutation and Randomization Tests
17.2.1 The Basic Construction
17.2.2 Asymptotic Results
17.3 Two-Sample Permutation Tests
17.4 Further Examples
17.5 Randomization Tests and Multiple Testing
17.6 Problems
17.7 Notes
18 Bootstrap and Subsampling Methods
18.1 Introduction
18.2 Basic Large-Sample Approximations
18.2.1 Pivotal Method
18.2.2 Asymptotic Pivotal Method
18.2.3 Asymptotic Approximation
18.3 Bootstrap Sampling Distributions
18.3.1 Introduction and Consistency
18.3.2 The Nonparametric Mean
18.3.3 Further Examples
18.4 Higher Order Asymptotic Comparisons
18.5 Hypothesis Testing
18.6 Stepdown Multiple Testing
18.7 Subsampling
18.7.1 The Basic Theorem in the I.I.D. Case
18.7.2 Comparison with the Bootstrap
18.7.3 Hypothesis Testing
18.8 Problems
18.9 Notes
Appendix A Auxiliary Results
A.1 Equivalence Relations; Groups
A.2 Convergence of Functions; Metric Spaces
A.3 Banach and Hilbert Spaces
A.4 Dominated Families of Distributions
A.5 The Weak Compactness Theorem
Appendix References
Subject Index
Author Index