Generalized radius processes for elliptically contoured distributions

• 成果类型: Article
• 署名作者: García-Escudero, LA; Gordaliza, A
• 署名单位: Universidad de Valladolid
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1198/016214504000002023
• 发表日期: 2005
• 页码: 1036-1045
• 关键词: multivariate; covariance; outliers; matrix

摘要

The use of Mahalanobis distances has a long history in statistics. Given a sample of size n and general location and scatter estimators, m(n) and Sigma(n), we can define generalized radii as r(i)(n) = root(X-i-m(n))' Sigma(-1)(n) (X-i-m(n)). If we wish to trim observations based on the estimators m(n) and Sigma(n), then it is natural to first remove the most remote ones (i.e., those with the largest r(i)(n,)s). With this in mind, we define a process that maps the trimming proportion, alpha in [0, 1], to the generalized radius of the observation that has just been removed by this level of trimming. We analyze the asymptotic behavior of this process for elliptically contoured distributions. We show that the limit law depends only on the elliptical family considered and how Sigma(n) serves to estimate the underlying scale factor through its determinant. We carry out Monte Carlo simulations for finite sample sizes, and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded.