Introduction to Statistical Limit Theory (Chapman & Hall/CRC Texts in Statistical Science v. 91)
By: Alan M. Polansky (author)Hardback
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Helping students develop a good understanding of asymptotic theory, Introduction to Statistical Limit Theory provides a thorough yet accessible treatment of common modes of convergence and their related tools used in statistics. It also discusses how the results can be applied to several common areas in the field. The author explains as much of the background material as possible and offers a comprehensive account of the modes of convergence of random variables, distributions, and moments, establishing a firm foundation for the applications that appear later in the book. The text includes detailed proofs that follow a logical progression of the central inferences of each result. It also presents in-depth explanations of the results and identifies important tools and techniques. Through numerous illustrative examples, the book shows how asymptotic theory offers deep insight into statistical problems, such as confidence intervals, hypothesis tests, and estimation. With an array of exercises and experiments in each chapter, this classroom-tested book gives students the mathematical foundation needed to understand asymptotic theory.
It covers the necessary introductory material as well as modern statistical applications, exploring how the underlying mathematical and statistical theories work together.
Alan M. Polansky is an associate professor in the Division of Statistics at Northern Illinois University. Dr. Polansky is the author of Observed Confidence Levels: Theory and Application (CRC Press, October 2007). His research interests encompass nonparametric statistics and industrial applications of statistics.
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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