# Testing Statistical Hypotheses

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1 E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer

2 Preface vii I Small-Sample Theory 1 1 The General Decision Problem Statistical Inference and Statistical Decisions Specification of a Decision Problem Randomization; Choice of Experiment Optimum Procedures Invariance and Unbiasedness ll 1.6 Bayes and Minimax Procedures Maximum Likelihood Iβ 1.8 Complete Classes Sufficient Statistics IS 1.10 Problems LM 1.1 I Notes 27 2 The Probability Background 28 2t Probability and Measure 2 S 2.2 Integration Statistics and Subfields Conditional Expectation and Probability Conditional Probability Distributions Characterization of Sufficiency Exponential Families 46

3 2.8 Problems 2.9 Notes Uniformly Most Powerful Tests Stating The Problem The Neyman-Pearson Fundamental Lemma p-values Distributions with Monotone Likelihood Ratio Confidence Bounds A Generalization of the Fundamental Lemma Two-Sided Hypotheses Least Favorable Distributions Applications to Normal Distributions Univariate Normal Models Multivariate Normal Models Problems Notes 107 Unbiasedness: Theory and First Applications Unbiasedness For Hypothesis Testing One-Parameter Exponential Families Ill 4.3 Similarity and Completeness UMP Unbiased Tests for Multiparameter Exponential Families Comparing Two Poisson or Binomial Populations Testing for Independence in a 2 x 2 Table Alternative Models for 2 x 2 Tables Some Three-Factor Contingency Tables The Sign Test Problems Notes 149 Unbiasedness: Applications to Normal Distributions Statistics Independent of a Sufficient Statistic Testing the Parameters of a Normal Distribution Comparing the Means and Variances of Two Normal Distributions Confidence Intervals and Families of Tests Unbiased Confidence Sets 1G4 5.0 Regression Bayesian Confidence Sets Permutation Tests Most Powerful Permutation Tests Randomization As A Basis For Inference Permutation Tests and Randomization Randomization Model and Confidence Intervals Testing for Independence in a Bivariate Normal Distribution Problems Notes 210

4 xi 6 Invariance Symmetry and Invariance Maximal Invariants Most Powerful Invariant Tests Sample Inspection by Variables Almost Invariance Unbiasedness and Invariance Admissibility Rank Tests The Two-Sample Problem The Hypothesis of Symmetry Equivariant Confidence Sets Average Smallest Equivariant Confidence Sets Confidence Bands for a Distribution Function Problems Notes Linear Hypotheses A Canonical Form Linear Hypotheses and Least Squares Tests of Homogeneity Two-Way Layout: One Observation per Cell Two-Way Layout: m Observations Per Cell Regression Random-Effects Model: One-way Classification Nested Classifications Multivariate Extensions Problems Notes The Minimax Principle Tests with Guaranteed Power Examples Comparing Two Approximate Hypotheses Maximin Tests and Invariance The Hunt Stein Theorem Most Stringent Tests Problems Notes Multiple Testing and Simultaneous Inference Introduction and the FWER Maximin Procedures The Hypothesis of Homogeneity Scheffé's 5-Method: A Special Case Scheffé's S-Method for General Linear Models Problems Notes 391

5 xii Contents 10 Conditional Inference Mixtures of Experiments Ancillary Statistics Optimal Conditional Tests Relevant Subsets Problems Notes 414 II Large-Sample Theory Basic Large Sample Theory Introduction Basic Convergence Concepts Weak Convergence and Central Limit Theorems Convergence in Probability and Applications Almost Sure Convergence Robustness of Some Classical Tests Effect of Distribution Effect of Dependence Robustness in Linear Models Nonparametric Mean Edgeworth Expansions The t-test A Result of Bahadur and Savage Alternative Tests Problems Notes Quadratic Mean Differentiable Families Introduction Quadratic Mean Differentiability (q.m.d.) Contiguity Likelihood Methods in Parametric Models Efficient Likelihood Estimation Wald Tests and Confidence Regions Rao Score Tests Likelihood Ratio Tests Problems Notes Large Sample Optimality Testing Sequences, Metrics, and Inequalities Asymptotic Relative Efficiency AUMP Tests in Univariate Models Asymptotically Normal Experiments Applications to Parametric Models One-sided Hypotheses Equivalence Hypotheses 559

6 xiii Multi-sided Hypotheses Applications to Nonparametric Models Nonparametric Mean Nonparametric Testing of Functionals Problems Notes Testing Goodness of Fit Introduction The Kolmogorov-Smirnov Test Simple Null Hypothesis Extensions of the Kolmogorov-Smirnov Test Pearson's Chi-squared Statistic Simple Null Hypothesis Chi-squared Test of Uniformity Composite Null Hypothesis Neyman's Smooth Tests Fixed k Asymptotics Neyman's Smooth Tests With Large k Weighted Quadratic Test Statistics Global Behavior of Power Functions Problems Notes General Large Sample Methods Introduction Permutation and Randomization Tests The Basic Construction Asymptotic Results Basic Large Sample Approximations Pivotal Method Asymptotic Pivotal Method Asymptotic Approximation Bootstrap Sampling Distributions Introduction and Consistency The Nonparametric Mean Further Examples Stepdown Multiple Testing Higher Order Asymptotic Comparisons Hypothesis Testing Subsampling The Basic Theorem in the I.I.D. Case Comparison with the Bootstrap Hypothesis Testing Problems Notes 690 A Auxiliary Results 692 A.l Equivalence Relations; Groups 592

7 xiv Contents A.2 Convergence of Functions; Metric Spaces 693 A.3 Banach and Hilbert Spaces 696 A.4 Dominated Families of Distributions 698 A.5 The Weak Compactness Theorem 700 References 702 Author Index 757 Subject Index 767

### Testing Statistical Hypotheses

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