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Examples and Problems in Mathematical Statistics

Book Description

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.

Table of Contents

  1. Cover
  2. Series
  3. Title Page
  4. Copyright Page
  5. Dedication
  6. Preface
  7. List of Random Variables
  8. List of Abbreviations
  9. Chapter 1: Basic Probability Theory
    1. PART I: THEORY
    2. 1.1 OPERATIONS ON SETS
    3. 1.2 ALGEBRA AND σ–FIELDS
    4. 1.3 PROBABILITY SPACES
    5. 1.4 CONDITIONAL PROBABILITIES AND INDEPENDENCE
    6. 1.5 RANDOM VARIABLES AND THEIR DISTRIBUTIONS
    7. 1.6 THE LEBESGUE AND STIELTJES INTEGRALS
    8. 1.7 JOINT DISTRIBUTIONS, CONDITIONAL DISTRIBUTIONS AND INDEPENDENCE
    9. 1.8 MOMENTS AND RELATED FUNCTIONALS
    10. 1.9 MODES OF CONVERGENCE
    11. 1.10 WEAK CONVERGENCE
    12. 1.11 LAWS OF LARGE NUMBERS
    13. 1.12 CENTRAL LIMIT THEOREM
    14. 1.13 MISCELLANEOUS RESULTS
    15. PART II: EXAMPLES
    16. PART III: PROBLEMS
    17. PART IV: SOLUTIONS TO SELECTED PROBLEMS
  10. Chapter 2: Statistical Distributions
    1. PART I: THEORY
    2. 2.1 INTRODUCTORY REMARKS
    3. 2.2 FAMILIES OF DISCRETE DISTRIBUTIONS
    4. 2.3 SOME FAMILIES OF CONTINUOUS DISTRIBUTIONS
    5. 2.4 TRANSFORMATIONS
    6. 2.5 VARIANCES AND COVARIANCES OF SAMPLE MOMENTS
    7. 2.6 DISCRETE MULTIVARIATE DISTRIBUTIONS
    8. 2.7 MULTINORMAL DISTRIBUTIONS
    9. 2.8 DISTRIBUTIONS OF SYMMETRIC QUADRATIC FORMS OF NORMAL VARIABLES
    10. 2.9 INDEPENDENCE OF LINEAR AND QUADRATIC FORMS OF NORMAL VARIABLES
    11. 2.10 THE ORDER STATISTICS
    12. 2.11 t–DISTRIBUTIONS
    13. 2.12 F–DISTRIBUTIONS
    14. 2.13 THE DISTRIBUTION OF THE SAMPLE CORRELATION
    15. 2.14 EXPONENTIAL TYPE FAMILIES
    16. 2.15 APPROXIMATING THE DISTRIBUTION OF THE SAMPLE MEAN: EDGEWORTH AND SADDLEPOINT APPROXIMATIONS
    17. PART II: EXAMPLES
    18. PART III: PROBLEMS
    19. PART IV: SOLUTIONS TO SELECTED PROBLEMS
  11. Chapter 3: Sufficient Statistics and the Information in Samples
    1. PART I: THEORY
    2. 3.1 INTRODUCTION
    3. 3.2 DEFINITION AND CHARACTERIZATION OF SUFFICIENT STATISTICS
    4. 3.3 LIKELIHOOD FUNCTIONS AND MINIMAL SUFFICIENT STATISTICS
    5. 3.4 SUFFICIENT STATISTICS AND EXPONENTIAL TYPE FAMILIES
    6. 3.5 SUFFICIENCY AND COMPLETENESS
    7. 3.6 SUFFICIENCY AND ANCILLARITY
    8. 3.7 INFORMATION FUNCTIONS AND SUFFICIENCY
    9. 3.8 THE FISHER INFORMATION MATRIX
    10. 3.9 SENSITIVITY TO CHANGES IN PARAMETERS
    11. PART II: EXAMPLES
    12. PART III: PROBLEMS
    13. PART IV: SOLUTIONS TO SELECTED PROBLEMS
  12. Chapter 4: Testing Statistical Hypotheses
    1. PART I: THEORY
    2. 4.1 THE GENERAL FRAMEWORK
    3. 4.2 THE NEYMAN–PEARSON FUNDAMENTAL LEMMA
    4. 4.3 TESTING ONE–SIDED COMPOSITE HYPOTHESES IN MLR MODELS
    5. 4.4 TESTING TWO–SIDED HYPOTHESES IN ONE–PARAMETER EXPONENTIAL FAMILIES
    6. 4.5 TESTING COMPOSITE HYPOTHESES WITH NUISANCE PARAMETERS—UNBIASED TESTS
    7. 4.6 LIKELIHOOD RATIO TESTS
    8. 4.7 THE ANALYSIS OF CONTINGENCY TABLES
    9. 4.8 SEQUENTIAL TESTING OF HYPOTHESES
    10. PART II: EXAMPLES
    11. PART III: PROBLEMS
    12. PART IV: SOLUTIONS TO SELECTED PROBLEMS
  13. Chapter 5: Statistical Estimation
    1. PART I: THEORY
    2. 5.1 GENERAL DISCUSSION
    3. 5.2 UNBIASED ESTIMATORS
    4. 5.3 THE EFFICIENCY OF UNBIASED ESTIMATORS IN REGULAR CASES
    5. 5.4 BEST LINEAR UNBIASED AND LEAST–SQUARES ESTIMATORS
    6. 5.5 STABILIZING THE LSE: RIDGE REGRESSIONS
    7. 5.6 MAXIMUM LIKELIHOOD ESTIMATORS
    8. 5.7 EQUIVARIANT ESTIMATORS
    9. 5.8 ESTIMATING EQUATIONS
    10. 5.9 PRETEST ESTIMATORS
    11. 5.10 ROBUST ESTIMATION OF THE LOCATION AND SCALE PARAMETERS OF SYMMETRIC DISTRIBUTIONS
    12. PART II: EXAMPLES
    13. PART III: PROBLEMS
    14. PART IV: SOLUTIONS OF SELECTED PROBLEMS
  14. Chapter 6: Confidence and Tolerance Intervals
    1. PART I: THEORY
    2. 6.1 GENERAL INTRODUCTION
    3. 6.2 THE CONSTRUCTION OF CONFIDENCE INTERVALS
    4. 6.3 OPTIMAL CONFIDENCE INTERVALS
    5. 6.4 TOLERANCE INTERVALS
    6. 6.5 DISTRIBUTION FREE CONFIDENCE AND TOLERANCE INTERVALS
    7. 6.6 SIMULTANEOUS CONFIDENCE INTERVALS
    8. 6.7 TWO–STAGE AND SEQUENTIAL SAMPLING FOR FIXED WIDTH CONFIDENCE INTERVALS
    9. PART II: EXAMPLES
    10. PART III: PROBLEMS
    11. PART IV: SOLUTION TO SELECTED PROBLEMS
  15. Chapter 7: Large Sample Theory for Estimation and Testing
    1. PART I: THEORY
    2. 7.1 CONSISTENCY OF ESTIMATORS AND TESTS
    3. 7.2 CONSISTENCY OF THE MLE
    4. 7.3 ASYMPTOTIC NORMALITY AND EFFICIENCY OF CONSISTENT ESTIMATORS
    5. 7.4 SECOND–ORDER EFFICIENCY OF BAN ESTIMATORS
    6. 7.5 LARGE SAMPLE CONFIDENCE INTERVALS
    7. 7.6 EDGEWORTH AND SADDLEPOINT APPROXIMATIONS TO THE DISTRIBUTION OF THE MLE: ONE–PARAMETER CANONICAL EXPONENTIAL FAMILIES
    8. 7.7 LARGE SAMPLE TESTS
    9. 7.8 PITMAN’S ASYMPTOTIC EFFICIENCY OF TESTS
    10. 7.9 ASYMPTOTIC PROPERTIES OF SAMPLE QUANTILES
    11. PART II: EXAMPLES
    12. PART III: PROBLEMS
    13. PART IV: SOLUTION OF SELECTED PROBLEMS
  16. Chapter 8: Bayesian Analysis in Testing and Estimation
    1. PART I: THEORY
    2. 8.1 THE BAYESIAN FRAMEWORK
    3. 8.2 BAYESIAN TESTING OF HYPOTHESIS
    4. 8.3 BAYESIAN CREDIBILITY AND PREDICTION INTERVALS
    5. 8.4 BAYESIAN ESTIMATION
    6. 8.5 APPROXIMATION METHODS
    7. 8.6 EMPIRICAL BAYES ESTIMATORS
    8. PART II: EXAMPLES
    9. PART III: PROBLEMS
    10. PART IV: SOLUTIONS OF SELECTED PROBLEMS
  17. Chapter 9: Advanced Topics in Estimation Theory
    1. PART I: THEORY
    2. 9.1 MINIMAX ESTIMATORS
    3. 9.2 MINIMUM RISK EQUIVARIANT, BAYES EQUIVARIANT, AND STRUCTURAL ESTIMATORS
    4. 9.3 THE ADMISSIBILITY OF ESTIMATORS
    5. PART II: EXAMPLES
    6. PART III: PROBLEMS
    7. PART IV: SOLUTIONS OF SELECTED PROBLEMS
  18. Reference
  19. Author Index
  20. Subject Index
  21. Wiley Series in Probability and Statistics