{"title":"Pseudo-maximization and self-normalized processes","authors":"V. Peña, M. Klass, T. Lai","doi":"10.1214/07-PS119","DOIUrl":null,"url":null,"abstract":"Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic inte- grals, martingale inequalities and limit theorems, likelihood-based methods in hypothesistesting and parameterestimation, and Studentizedpivots and bootstrap-t methods for confidence intervals. In contrast to standard nor- malization, large values of the observationsplay a lesser role as they appear both in the numerator and its self-normalized denominator, thereby mak- ing the process scale invariantand contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of de- pendentvariablesand 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.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2007-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/07-PS119","citationCount":"37","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability Surveys","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/07-PS119","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 37
Abstract
Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic inte- grals, martingale inequalities and limit theorems, likelihood-based methods in hypothesistesting and parameterestimation, and Studentizedpivots and bootstrap-t methods for confidence intervals. In contrast to standard nor- malization, large values of the observationsplay a lesser role as they appear both in the numerator and its self-normalized denominator, thereby mak- ing the process scale invariantand contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of de- pendentvariablesand 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.