Contents

Cover

Series

Title Page

Copyright Page

Dedication

Preface

List of Random Variables

List of Abbreviations

Chapter 1: Basic Probability Theory

PART I: THEORY

1.1 OPERATIONS ON SETS

1.2 ALGEBRA AND σ–FIELDS

1.3 PROBABILITY SPACES

1.4 CONDITIONAL PROBABILITIES AND INDEPENDENCE

1.5 RANDOM VARIABLES AND THEIR DISTRIBUTIONS

1.6 THE LEBESGUE AND STIELTJES INTEGRALS

1.7 JOINT DISTRIBUTIONS, CONDITIONAL DISTRIBUTIONS AND INDEPENDENCE

1.8 MOMENTS AND RELATED FUNCTIONALS

1.9 MODES OF CONVERGENCE

1.10 WEAK CONVERGENCE

1.11 LAWS OF LARGE NUMBERS

1.12 CENTRAL LIMIT THEOREM

1.13 MISCELLANEOUS RESULTS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 2: Statistical Distributions

PART I: THEORY

2.1 INTRODUCTORY REMARKS

2.2 FAMILIES OF DISCRETE DISTRIBUTIONS

2.3 SOME FAMILIES OF CONTINUOUS DISTRIBUTIONS

2.4 TRANSFORMATIONS

2.5 VARIANCES AND COVARIANCES OF SAMPLE MOMENTS

2.6 DISCRETE MULTIVARIATE DISTRIBUTIONS

2.7 MULTINORMAL DISTRIBUTIONS

2.8 DISTRIBUTIONS OF SYMMETRIC QUADRATIC FORMS OF NORMAL VARIABLES

2.9 INDEPENDENCE OF LINEAR AND QUADRATIC FORMS OF NORMAL VARIABLES

2.10 THE ORDER STATISTICS

2.11 t–DISTRIBUTIONS

2.12 F–DISTRIBUTIONS

2.13 THE DISTRIBUTION OF THE SAMPLE CORRELATION

2.14 EXPONENTIAL TYPE FAMILIES

2.15 APPROXIMATING THE DISTRIBUTION OF THE SAMPLE MEAN: EDGEWORTH AND SADDLEPOINT APPROXIMATIONS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 3: Sufficient Statistics and the Information in Samples

PART I: THEORY

3.1 INTRODUCTION

3.2 DEFINITION AND CHARACTERIZATION OF SUFFICIENT STATISTICS

3.3 LIKELIHOOD FUNCTIONS AND MINIMAL SUFFICIENT STATISTICS

3.4 SUFFICIENT STATISTICS AND EXPONENTIAL TYPE FAMILIES

3.5 SUFFICIENCY AND COMPLETENESS

3.6 SUFFICIENCY AND ANCILLARITY

3.7 INFORMATION FUNCTIONS AND SUFFICIENCY

3.8 THE FISHER INFORMATION MATRIX

3.9 SENSITIVITY TO CHANGES IN PARAMETERS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 4: Testing Statistical Hypotheses

PART I: THEORY

4.1 THE GENERAL FRAMEWORK

4.2 THE NEYMAN–PEARSON FUNDAMENTAL LEMMA

4.3 TESTING ONE–SIDED COMPOSITE HYPOTHESES IN MLR MODELS

4.4 TESTING TWO–SIDED HYPOTHESES IN ONE–PARAMETER EXPONENTIAL FAMILIES

4.5 TESTING COMPOSITE HYPOTHESES WITH NUISANCE PARAMETERS—UNBIASED TESTS

4.6 LIKELIHOOD RATIO TESTS

4.7 THE ANALYSIS OF CONTINGENCY TABLES

4.8 SEQUENTIAL TESTING OF HYPOTHESES

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS TO SELECTED PROBLEMS

Chapter 5: Statistical Estimation

PART I: THEORY

5.1 GENERAL DISCUSSION

5.2 UNBIASED ESTIMATORS

5.3 THE EFFICIENCY OF UNBIASED ESTIMATORS IN REGULAR CASES

5.4 BEST LINEAR UNBIASED AND LEAST–SQUARES ESTIMATORS

5.5 STABILIZING THE LSE: RIDGE REGRESSIONS

5.6 MAXIMUM LIKELIHOOD ESTIMATORS

5.7 EQUIVARIANT ESTIMATORS

5.8 ESTIMATING EQUATIONS

5.9 PRETEST ESTIMATORS

5.10 ROBUST ESTIMATION OF THE LOCATION AND SCALE PARAMETERS OF SYMMETRIC DISTRIBUTIONS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS OF SELECTED PROBLEMS

Chapter 6: Confidence and Tolerance Intervals

PART I: THEORY

6.1 GENERAL INTRODUCTION

6.2 THE CONSTRUCTION OF CONFIDENCE INTERVALS

6.3 OPTIMAL CONFIDENCE INTERVALS

6.4 TOLERANCE INTERVALS

6.5 DISTRIBUTION FREE CONFIDENCE AND TOLERANCE INTERVALS

6.6 SIMULTANEOUS CONFIDENCE INTERVALS

6.7 TWO–STAGE AND SEQUENTIAL SAMPLING FOR FIXED WIDTH CONFIDENCE INTERVALS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTION TO SELECTED PROBLEMS

Chapter 7: Large Sample Theory for Estimation and Testing

PART I: THEORY

7.1 CONSISTENCY OF ESTIMATORS AND TESTS

7.2 CONSISTENCY OF THE MLE

7.3 ASYMPTOTIC NORMALITY AND EFFICIENCY OF CONSISTENT ESTIMATORS

7.4 SECOND–ORDER EFFICIENCY OF BAN ESTIMATORS

7.5 LARGE SAMPLE CONFIDENCE INTERVALS

7.6 EDGEWORTH AND SADDLEPOINT APPROXIMATIONS TO THE DISTRIBUTION OF THE MLE: ONE–PARAMETER CANONICAL EXPONENTIAL FAMILIES

7.7 LARGE SAMPLE TESTS

7.8 PITMAN’S ASYMPTOTIC EFFICIENCY OF TESTS

7.9 ASYMPTOTIC PROPERTIES OF SAMPLE QUANTILES

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTION OF SELECTED PROBLEMS

Chapter 8: Bayesian Analysis in Testing and Estimation

PART I: THEORY

8.1 THE BAYESIAN FRAMEWORK

8.2 BAYESIAN TESTING OF HYPOTHESIS

8.3 BAYESIAN CREDIBILITY AND PREDICTION INTERVALS

8.4 BAYESIAN ESTIMATION

8.5 APPROXIMATION METHODS

8.6 EMPIRICAL BAYES ESTIMATORS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS OF SELECTED PROBLEMS

Chapter 9: Advanced Topics in Estimation Theory

PART I: THEORY

9.1 MINIMAX ESTIMATORS

9.2 MINIMUM RISK EQUIVARIANT, BAYES EQUIVARIANT, AND STRUCTURAL ESTIMATORS

9.3 THE ADMISSIBILITY OF ESTIMATORS

PART II: EXAMPLES

PART III: PROBLEMS

PART IV: SOLUTIONS OF SELECTED PROBLEMS

Reference

Author Index

Subject Index

Wiley Series in Probability and Statistics