Examples and Problems in Mathematical Statistics

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WILEY SERIES IN PROBABILITY AND STATISTICS
Examples and Problems in Mathematical Statistics
Copyright
     © 2014 by John Wiley & Sons
     ISBN 978-1-118-60550-9
     QC32.Z265 2013 519.5–dc23
     LCCN 2013034492
Dedication
Contents
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 s-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
          1.6.1 General Definition of Expected Value: The Lebesgue Integral
          1.6.2 The Stieltjes–Riemann Integral
          1.6.3 Mixtures of Discrete and Absolutely Continuous Distributions
          1.6.4 Quantiles of Distributions
          1.6.5 Transformations
     1.7 JOINT DISTRIBUTIONS, CONDITIONAL DISTRIBUTIONS AND INDEPENDENCE
          1.7.1 Joint Distributions
          1.7.2 Conditional Expectations: General Definition
          1.7.3 Independence
     1.8 MOMENTS AND RELATED FUNCTIONALS
     1.9 MODES OF CONVERGENCE
     1.10 WEAK CONVERGENCE
     1.11 LAWS OF LARGE NUMBERS
          1.11.1 The Weak Law of Large Numbers (WLLN)
          1.11.2 The Strong Law of Large Numbers (SLLN)
     1.12 CENTRAL LIMIT THEOREM
     1.13 MISCELLANEOUS RESULTS
          1.13.1 Law of the Iterated Logarithm
          1.13.2 Uniform Integrability
          1.13.3 Inequalities
          1.13.4 The Delta Method
          1.13.5 The Symbols op and Op
          1.13.6 The Empirical Distribution and Sample Quantiles
     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
          2.2.1 Binomial Distributions
          2.2.2 Hypergeometric Distributions
          2.2.3 Poisson Distributions
          2.2.4 Geometric, Pascal, and Negative Binomial Distributions
     2.3 SOME FAMILIES OF CONTINUOUS DISTRIBUTIONS
          2.3.1 Rectangular Distributions
          2.3.2 Beta Distributions
          2.3.3 Gamma Distributions
          2.3.4 Weibull and Extreme Value Distributions
          2.3.5 Normal Distributions
          2.3.6 Normal Approximations
     2.4 TRANSFORMATIONS
          2.4.1 One-to-One Transformations of Several Variables
          2.4.2 Distribution of Sums
          2.4.3 Distribution of Ratios
     2.5 VARIANCES AND COVARIANCES OF SAMPLE MOMENTS
     2.6 DISCRETE MULTIVARIATE DISTRIBUTIONS
          2.6.1 The Multinomial Distribution
          2.6.2 Multivariate Negative Binomial
          2.6.3 Multivariate Hypergeometric Distributions
     2.7 MULTINORMAL DISTRIBUTIONS
          2.7.1 Basic Theory
          2.7.2 Distribution of Subvectors and Distributions of Linear Forms
          2.7.3 Independence of Linear Forms
     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
          2.15.1 Edgeworth Expansion
          2.15.2 Saddlepoint Approximation
     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
          3.2.1 Introductory Discussion
          3.2.2 Theoretical Formulation
     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
          3.7.1 The Fisher Information
          3.7.2 The Kullback–Leibler Information
     3.8 THE FISHER INFORMATION MATRIX
     3.9 SENSITIVITY TO CHANGES IN PARAMETERS
          3.9.1 The Hellinger Distance
     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
          4.6.1 Testing in Normal Regression Theory
          4.6.2 Comparison of Normal Means: The Analysis of Variance
     4.7 THE ANALYSIS OF CONTINGENCY TABLES
          4.7.1 The Structure of Multi-Way Contingency Tables and the Statistical Model
          4.7.2 Testing the Significance of Association
          4.7.3 The Analysis of  Tables
          4.7.4 Likelihood Ratio Tests for Categorical Data
     4.8 SEQUENTIAL TESTING OF HYPOTHESES
          4.8.1 The Wald Sequential Probability Ratio Test
     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
          5.2.1 General Definition and Example
          5.2.2 Minimum Variance Unbiased Estimators
          5.2.3 The Cramér–Rao Lower Bound for the One-Parameter Case
          5.2.4 Extension of the Cramér–Rao Inequality to Multiparameter Cases
          5.2.5 General Inequalities of the Cramér–Rao Type
     5.3 THE EFFICIENCY OF UNBIASED ESTIMATORS IN REGULAR CASES
     5.4 BEST LINEAR UNBIASED AND LEAST-SQUARES ESTIMATORS
          5.4.1 BLUEs of the Mean
          5.4.2 Least-Squares and BLUEs in Linear Models
          5.4.3 Best Linear Combinations of Order Statistics
     5.5 STABILIZING THE LSE: RIDGE REGRESSIONS
     5.6 MAXIMUM LIKELIHOOD ESTIMATORS
          5.6.1 Definition and Examples
          5.6.2 MLEs in Exponential Type Families
          5.6.3 The Invariance Principle
          5.6.4 MLE of the Parameters of Tolerance Distributions
     5.7 EQUIVARIANT ESTIMATORS
          5.7.1 The Structure of Equivariant Estimators
          5.7.2 Minimum MSE Equivariant Estimators
          5.7.3 Minimum Risk Equivariant Estimators
          5.7.4 The Pitman Estimators
     5.8 ESTIMATING EQUATIONS
          5.8.1 Moment-Equations Estimators
          5.8.2 General Theory of Estimating Functions
     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
          8.1.1 Prior, Posterior, and Predictive Distributions
          8.1.2 Noninformative and Improper Prior Distributions
          8.1.3 Risk Functions and Bayes Procedures
     8.2 BAYESIAN TESTING OF HYPOTHESIS
          8.2.1 Testing Simple Hypothesis
          8.2.2 Testing Composite Hypotheses
          8.2.3 Bayes Sequential Testing of Hypotheses
     8.3 BAYESIAN CREDIBILITY AND PREDICTION INTERVALS
          8.3.1 Credibility Intervals
          8.3.2 Prediction Intervals
     8.4 BAYESIAN ESTIMATION
          8.4.1 General Discussion and Examples
          8.4.2 Hierarchical Models
          8.4.3 The Normal Dynamic Linear Model
     8.5 APPROXIMATION METHODS
          8.5.1 Analytical Approximations
          8.5.2 Numerical Approximations
     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
          9.2.1 Formal Bayes Estimators for Invariant Priors
          9.2.2 Equivariant Estimators Based on Structural Distributions
     9.3 THE ADMISSIBILITY OF ESTIMATORS
          9.3.1 Some Basic Results
          9.3.2 The Inadmissibility of Some Commonly Used Estimators
          9.3.3 Minimax and Admissible Estimators of the Location Parameter
          9.3.4 The Relationship of Empirical Bayes and Stein-Type Estimators of the Location Parameter in the Normal Case
     PART II: EXAMPLES
     PART III: PROBLEMS
     PART IV: SOLUTIONS OF SELECTED PROBLEMS
References
Author Index
Subject Index
List of Published Books of WILEY SERIES IN PROBABILITY AND STATISTICS