Pseudo-maximization and self-normalized processes

IF 1.3 Q2 STATISTICS & PROBABILITY Probability Surveys Pub Date : 2007-09-14 DOI:10.1214/07-PS119
V. Peña, M. Klass, T. Lai
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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.
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伪最大化和自规范化过程
自归一化过程是许多概率和统计研究的基础。它们自然地出现在随机积分、鞅不等式和极限定理、假设检验和参数估计中的基于似然的方法以及置信区间的学生枢轴和bootstrap方法的研究中。与标准非规范化相比,观测值的大值表现出较小的作用,因为它们同时出现在分子和其自归一化的分母中,从而使过程规模不变并有助于其鲁棒性。在此,我们调查了自归一化过程在相依变量情况下的一些结果,并描述了一种称为“伪最大化”的关键方法,该方法已被用于推导这些结果。在多元情况下,自归一化由乘以一个正定矩阵的逆组成(而不是像在标量情况下除以一个正随机变量),并且在统计应用中无处不在,给出了其示例。AMS 2000学科分类:初级60K35、初级60K35;二次60 k35。
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
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