Those moving on to advanced statistics typically lack the mathematical foundation that allows them to make full use of statistical limit theory. This accessible resource reviews approximation theory and limit theory for sequences of functions and basic notions of functional analysis. It provides detailed arguments that show how underlying mathematical and statistical theory work together. Among its unique qualities, the text covers expansion theory, which is becoming increasingly important in modern applications. It also discusses bootstrap, kernel smoothing, and Markov chain Monte Carlo and includes a wide array of examples and problems from the fundamental to very advanced.
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 rth 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.
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