Introduction to Statistical Limit Theory (Hardcover)

Sequences of Real Numbers and Functions
Introduction
Sequences of Real Numbers
Sequences of Real Functions
The Taylor Expansion
Asymptotic Expansions
Inversion of Asymptotic Expansions

Random Variables and Characteristic Functions
Introduction
Probability Measures and Random Variables
Some Important Inequalities
Some Limit Theory for Events
Generating and Characteristic Functions

Convergence of Random Variables
Introduction
Convergence in Probability
Stronger Modes of Convergence
Convergence of Random Vectors
Continuous Mapping Theorems
Laws of Large Numbers
The Glivenko?Cantelli Theorem
Sample Moments
Sample Quantiles

Convergence of Distributions
Introduction
Weak Convergence of Random Variables
Weak Convergence of Random Vectors
The Central Limit Theorem
The Accuracy of the Normal Approximation
The Sample Moments
The Sample Quantiles

Convergence of Moments
Convergence in r^th Mean
Uniform Integrability
Convergence of Moments

Central Limit Theorems
Introduction
Non-Identically Distributed Random Variables
Triangular Arrays
Transformed Random Variables

Asymptotic Expansions for Distributions
Approximating a Distribution
Edgeworth Expansions
The Cornish?Fisher Expansion
The Smooth Function Model
General Edgeworth and Cornish?Fisher Expansions
Studentized Statistics
Saddlepoint Expansions

Asymptotic Expansions for Random Variables
Approximating Random Variables
Stochastic Order Notation
The Delta Method
The Sample Moments

Differentiable Statistical Functionals
Introduction
Functional Parameters and Statistics
Differentiation of Statistical Functionals
Expansion Theory for Statistical Functionals
Asymptotic Distribution

Parametric Inference
Introduction
Point Estimation
Confidence Intervals
Statistical Hypothesis Tests
Observed Confidence Levels
Bayesian Estimation

Nonparametric Inference
Introduction
Unbiased Estimation and U-Statistics
Linear Rank Statistics
Pitman Asymptotic Relative Efficiency
Density Estimation
The Bootstrap

Appendix A: Useful Theorems and Notation
Appendix B: Using R for Experimentation

References

Exercises and Experiments appear at the end of each chapter.