Self-Normalized Processes Limit Theory and Statistical Applications

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.

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.

Victor H. de la Peña is Fellow of Institute of Mathematical Statistics and a Medallion Lecturer for IMS in 2007.

Tze Leung LAI: Distinguished Lecture Series in Statistical Science from Academia Sinica (2001), Starr Lectures in Financial Mathematics from the University of Hong Kong (2001), Center for Advanced Study in the Behavioral Sciences Fellowship (1999-2000), Richard Anderson Lecture in Statistics from University of Kentucky (1999), Election to Academia Sinica (1994), Committee of Presidents of Statistical Societies Award (1983), John Simon Guggenheim Fellowship (1983-84).

Qi-Man SHAO is Associate Editor of 5 top journals and co-author of: Chen, M. H., Shao, Q. M. and Ibrahim, J.G. (2000) , Monte Carlo Methods In Bayesian Computation . Springer Series in Statistics, Springer-Verlag , New York. ISBN 0-387-98935-8