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Dedication
by Shelemyahu Zacks
Examples and Problems in Mathematical Statistics
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
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Preface
To my wife Hanna,
our sons Yuval and David,
and their families, with love.
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