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 Probability Surveys > Vol. 4 (2007) open journal systems 

Pseudo-maximization and self-normalized processes

Victor H. de la Peña, Columbia University
Michael J. Klass, UC Berkeley
Tze-Leung Lai, Stanford University

Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and parameter estimation, and Studentized pivots and bootstrap-t methods for confidence intervals. In contrast to standard normalization, large values of the observations play a lesser role as they appear both in the numerator and its self-normalized denominator, thereby making the process scale invariant and contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of dependent variables and describe a key method called ``pseudo-maximization'' that has been used to derive these results. In the multivariate case, self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case) and is ubiquitous in statistical applications, examples of which are given.

AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35.

Keywords: Self-normalization, method of mixtures, moment and exponential inequalities, LIL.

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Peña, Victor H. De La, Klass, Michael J., Lai, Tze-Leung, Pseudo-maximization and self-normalized processes, Probability Surveys, 4, (2007), 172-192 (electronic). DOI: 10.1214/07-PS119.


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Probability Surveys. ISSN: 1549-5787