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arXiv · 0709.2233

Pseudo-maximization and self-normalized processes

Victor H. de la Peña·Michael J. Klass·Tze Leung Lai
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Abstract

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. 2007-10-10. Pseudo-maximization and self-normalized processes. https://doi.org/10.1214/07-ps119

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