Provides the necessary skills to solve problems in mathematical statistics through theory, concrete examples, and exercises
With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplication and presents numerous problem-solving examples that illustrate the relatednotations and proven results.
Written by an established authority in probability and mathematical statistics, each chapter begins with a theoretical presentation to introduce both the topic and the important results in an effort to aid in overall comprehension. Examples are then provided, followed by problems, and finally, solutions to some of the earlier problems. In addition, Examples and Problems in Mathematical Statistics features:
Over 160 practical and interesting real-world examples from a variety of fields including engineering, mathematics, and statistics to help readers become proficient in theoretical problem solving
More than 430 unique exercises with select solutions
Key statistical inference topics, such as probability theory, statistical distributions, sufficient statistics, information in samples, testing statistical hypotheses, statistical estimation, confidence and tolerance intervals, large sample theory, and Bayesian analysis
Recommended for graduate-level courses in probability and statistical inference, Examples and Problems in Mathematical Statistics is also an ideal reference for applied statisticians and researchers.
Author(s): Shelemyahu Zacks
Edition: 1
Publisher: Wiley
Year: 2014
Language: English
Pages: C, xx, 622
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;
Cover
S Title
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
