Pseudo-maximization and self-normalized processes

TL;DR

This paper surveys self-normalized processes, highlighting their robustness and scale invariance, and introduces the pseudo-maximization method used to derive key probabilistic and statistical results, especially in dependent and multivariate cases.

Contribution

It provides a comprehensive overview of self-normalized processes and details the pseudo-maximization technique for analyzing these processes in dependent and multivariate contexts.

Findings

Self-normalized processes are scale invariant and robust.

Pseudo-maximization is a key method for deriving results.

Applications include martingale inequalities and bootstrap methods.

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.