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Mathematics - Probability Theory and Stochastic Processes | Self-Normalized Processes - Limit Theory and Statistical Applications

Self-Normalized Processes

Limit Theory and Statistical Applications

Peña, Victor H., Lai, Tze Leung, Shao, Qi-Man

2009, XIV, 275 p.

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  • Systematically treats the theory and applications of self-normalization
  • Fills a current gap in PhD level courses in probability and statistics offered by major Statistics departments
  • Rich enough in its coverage to provide such a second course for PhD students
  • Integrates advanced probability with theoretical statistics, instead of presenting them as two disparate subjects
  • Provides PhD students important tools for their thesis research if they should work on statistical theory

Self-normalized processes are of common occurrence in probabilistic and statistical studies. A prototypical example is Student's t-statistic introduced in 1908 by Gosset, whose portrait is on the front cover. Due to the highly non-linear nature of these processes, the theory experienced a long period of slow development. In recent years there have been a number of important advances in the theory and applications of self-normalized processes. Some of these developments are closely linked to the study of central limit theorems, which imply that self-normalized processes are approximate pivots for statistical inference.

The present volume covers recent developments in the area, including self-normalized large and moderate deviations, and laws of the iterated logarithms for self-normalized martingales. This is the first book that systematically treats the theory and applications of self-normalization.

Content Level » Research

Keywords » Bootstrapping - Likelihood - Random variable - bootstrap - calculus - large and moderate deviations - law of the iterated logarithm - self-normalization - sequential analysis - studentized U-statistic - t-statistic

Related subjects » Probability Theory and Stochastic Processes - Statistical Theory and Methods

Table of contents / Sample pages 

Independent Random Variables.- Classical Limit Theorems, Inequalities and Other Tools.- Self-Normalized Large Deviations.- Weak Convergence of Self-Normalized Sums.- Stein's Method and Self-Normalized Berry–Esseen Inequality.- Self-Normalized Moderate Deviations and Laws of the Iterated Logarithm.- Cramér-Type Moderate Deviations for Self-Normalized Sums.- Self-Normalized Empirical Processes and U-Statistics.- Martingales and Dependent Random Vectors.- Martingale Inequalities and Related Tools.- A General Framework for Self-Normalization.- Pseudo-Maximization via Method of Mixtures.- Moment and Exponential Inequalities for Self-Normalized Processes.- Laws of the Iterated Logarithm for Self-Normalized Processes.- Multivariate Self-Normalized Processes with Matrix Normalization.- Statistical Applications.- The t-Statistic and Studentized Statistics.- Self-Normalization for Approximate Pivots in Bootstrapping.- Pseudo-Maximization in Likelihood and Bayesian Inference.- Sequential Analysis and Boundary Crossing Probabilities for Self-Normalized Statistics.

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