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Asymptotic Statistics

Book Description

This book is an introduction to the field of asymptotic statistics. The treatment is both practical and mathematically rigorous. In addition to most of the standard topics of an asymptotics course, including likelihood inference, M-estimation, the theory of asymptotic efficiency, U-statistics, and rank procedures, the book also presents recent research topics such as semiparametric models, the bootstrap, and empirical processes and their applications. The topics are organized from the central idea of approximation by limit experiments, which gives the book one of its unifying themes. This entails mainly the local approximation of the classical i.i.d. set up with smooth parameters by location experiments involving a single, normally distributed observation. Thus, even the standard subjects of asymptotic statistics are presented in a novel way. Suitable as a graduate or Master's level statistics text, this book will also give researchers an overview of research in asymptotic statistics.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication
  5. Contents
  6. Preface
  7. Notation
  8. 1. Introduction
    1. 1.1. Approximate Statistical Procedures
    2. 1.2. Asymptotic Optimality Theory
    3. 1.3. Limitations
    4. 1.4. The Index n
  9. 2. Stochastic Convergence
    1. 2.1. Basic Theory
    2. 2.2. Stochastic o and O Symbols
    3. *2.3. Characteristic Functions
    4. *2.4. Almost-Sure Representations
    5. *2.5. Convergence of Moments
    6. *2.6. Convergence-Determining Classes
    7. *2.7. Law of the Iterated Logarithm
    8. *2.8. Lindeberg-Feller Theorem
    9. *2.9. Convergence in Total Variation
    10. Problems
  10. 3. Delta Method
    1. 3.1. Basic Result
    2. 3.2. Variance-Stabilizing Transformations
    3. *3.3. Higher-Order Expansions
    4. *3.4. Uniform Delta Method
    5. *3.5. Moments
    6. Problems
  11. 4. Moment Estimators
    1. 4.1. Method of Moments
    2. *4.2. Exponential Families
    3. Problems
  12. 5. M- and Z-Estimators
    1. 5.1. Introduction
    2. 5.2. Consistency
    3. 5.3. Asymptotic Normality
    4. *5.4. Estimated Parameters
    5. 5.5. Maximum Likelihood Estimators
    6. *5.6. Classical Conditions
    7. *5.7. One-Step Estimators
    8. *5.8. Rates of Convergence
    9. *5.9. Argmax Theorem
    10. Problems
  13. 6. Contiguity
    1. 6.1. Likelihood Ratios
    2. 6.2. Contiguity
    3. Problems
  14. 7. Local Asymptotic Normality
    1. 7.1. Introduction
    2. 7.2. Expanding the Likelihood
    3. 7.3. Convergence to a Normal Experiment
    4. 7.4. Maximum Likelihood
    5. *7.5. Limit Distributions under Alternatives
    6. *7.6. Local Asymptotic Normality
    7. Problems
  15. 8. Efficiency of Estimators
    1. 8.1. Asymptotic Concentration
    2. 8.2. Relative Efficiency
    3. 8.3. Lower Bound for Experiments
    4. 8.4. Estimating Normal Means
    5. 8.5. Convolution Theorem
    6. 8.6. Almost-Everywhere Convolution Theorem
    7. *8.7. Local Asymptotic Minimax Theorem
    8. *8.8. Shrinkage Estimators
    9. *8.9. Achieving the Bound
    10. *8.10. Large Deviations
    11. Problems
  16. 9. Limits of Experiments
    1. 9.1. Introduction
    2. 9.2. Asymptotic Representation Theorem
    3. 9.3. Asymptotic Normality
    4. 9.4. Uniform Distribution
    5. 9.5. Pareto Distribution
    6. 9.6. Asymptotic Mixed Normality
    7. 9.7. Heuristics
    8. Problems
  17. 10. Bayes Procedures
    1. 10.1. Introduction
    2. 10.2. Bernstein–von Mises Theorem
    3. 10.3. Point Estimators
    4. *10.4. Consistency
    5. Problems
  18. 11. Projections
    1. 11.1. Projections
    2. 11.2. Conditional Expectation
    3. 11.3. Projection onto Sums
    4. *11.4. Hoeffding Decomposition
    5. Problems
  19. 12. U-Statistics
    1. 12.1. One-Sample U-Statistics
    2. 12.2. Two-Sample U-statistics
    3. *12.3. Degenerate U-Statistics
    4. Problems
  20. 13. Rank, Sign, and Permutation Statistics
    1. 13.1. Rank Statistics
    2. 13.2. Signed Rank Statistics
    3. 13.3. Rank Statistics for Independence
    4. *13.4. Rank Statistics under Alternatives
    5. 13.5. Permutation Tests
    6. *13.6. Rank Central Limit Theorem
    7. Problems
  21. 14. Relative Efficiency of Tests
    1. 14.1. Asymptotic Power Functions
    2. 14.2. Consistency
    3. 14.3. Asymptotic Relative Efficiency
    4. *14.4. Other Relative Efficiencies
    5. *14.5. Rescaling Rates
    6. Problems
  22. 15. Efficiency of Tests
    1. 15.1. Asymptotic Representation Theorem
    2. 15.2. Testing Normal Means
    3. 15.3. Local Asymptotic Normality
    4. 15.4. One-Sample Location
    5. 15.5. Two-Sample Problems
    6. Problems
  23. 16. Likelihood Ratio Tests
    1. 16.1. Introduction
    2. *16.2. Taylor Expansion
    3. 16.3. Using Local Asymptotic Normality
    4. 16.4. Asymptotic Power Functions
    5. 16.5. Bartlett Correction
    6. *16.6. Bahadur Efficiency
    7. Problems
  24. 17. Chi-Square Tests
    1. 17.1. Quadratic Forms in Normal Vectors
    2. 17.2. Pearson Statistic
    3. 17.3. Estimated Parameters
    4. 17.4. Testing Independence
    5. *17.5. Goodness-of-Fit Tests
    6. *17.6. Asymptotic Efficiency
    7. Problems
  25. 18. Stochastic Convergence in Metric Spaces
    1. 18.1. Metric and Normed Spaces
    2. 18.2. Basic Properties
    3. 18.3. Bounded Stochastic Processes
    4. Problems
  26. 19. Empirical Processes
    1. 19.1. Empirical Distribution Functions
    2. 19.2. Empirical Distributions
    3. 19.3. Goodness-of-Fit Statistics
    4. 19.4. Random Functions
    5. 19.5. Changing Classes
    6. 19.6. Maximal Inequalities
    7. Problems
  27. 20. Functional Delta Method
    1. 20.1. von Mises Calculus
    2. 20.2. Hadamard-Differentiable Functions
    3. 20.3. Some Examples
    4. Problems
  28. 21. Quantiles and Order Statistics
    1. 21.1. Weak Consistency
    2. 21.2. Asymptotic Normality
    3. 21.3. Median Absolute Deviation
    4. 21.4. Extreme Values
    5. Problems
  29. 22. L-Statistics
    1. 22.1. Introduction
    2. 22.2. Hájek Projection
    3. 22.3. Delta Method
    4. 22.4. L-Estimators for Location
    5. Problems
  30. 23. Bootstrap
    1. 23.1. Introduction
    2. 23.2. Consistency
    3. 23.3. Higher-Order Correctness
    4. Problems
  31. 24. Nonparametric Density Estimation
    1. 24.1. Introduction
    2. 24.2. Kernel Estimators
    3. 24.3. Rate Optimality
    4. 24.4. Estimating a Unimodal Density
    5. Problems
  32. 25. Semiparametric Models
    1. 25.1. Introduction
    2. 25.2. Banach and Hilbert Spaces
    3. 25.3. Tangent Spaces and Information
    4. 25.4. Efficient Score Functions
    5. 25.5. Score and Information Operators
    6. 25.6. Testing
    7. *25.7. Efficiency and the Delta Method
    8. 25.8. Efficient Score Equations
    9. 25.9. General Estimating Equations
    10. 25.10. Maximum Likelihood Estimators
    11. 25.11. Approximately Least-Favorable Submodels
    12. Problems
  33. References
  34. Index