Open Access
2007 Pseudo-maximization and self-normalized processes
Victor H. de la Peña, Michael J. Klass, Tze Leung Lai
Probab. Surveys 4: 172-192 (2007). DOI: 10.1214/07-PS119


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.


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Victor H. de la Peña. Michael J. Klass. Tze Leung Lai. "Pseudo-maximization and self-normalized processes." Probab. Surveys 4 172 - 192, 2007.


Published: 2007
First available in Project Euclid: 11 October 2007

zbMATH: 1189.60057
MathSciNet: MR2368950
Digital Object Identifier: 10.1214/07-PS119

Primary: 60K35 , 60K35
Secondary: 60K35

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

Rights: Copyright © 2007 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.4 • 2007
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