Generalized Radius Processes for Elliptically Contoured Distributions

• Published in: Journal of the American Statistical Association Vol. 100; no. 471; pp. 1036 - 1045
• Main Authors: García-Escudero, Luis Angel, Gordaliza, Alfonso
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.09.2005; American Statistical Association; Taylor & Francis Ltd
• Subjects: Approximation; Asymptotic; Brownian bridge; Covariance; Covariance matrices; Critical values; Distribution theory; Estimators; Exact sciences and technology; Gaussian distributions; Gaussian process; Influence function; Mahalanobis distance; Mathematics; Minimum covariance determinant estimation; Monte Carlo simulation; Multivariate analysis; Outliers; Parameter estimation; Probability and statistics; Probability theory and stochastic processes; Robustness; Sample size; Sampling distributions; Sampling theory, sample surveys; Sciences and techniques of general use; Statistics; Theory and Methods; Minimum covariance determinant estimation; Monte Carlo method; Covariance analysis; Asymptotic behavior; Finite sample; Asymptotic; Statistical method; Influence function; Gaussian process; Limit distribution; Gaussian processes; Distribution function; Robustness; Mahalanobis distance
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214504000002023

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 Σ n , we can define "generalized" radii as . If we wish to trim observations based on the estimators m n and Σ n , then it is natural to first remove the most remote ones (i.e., those with the largest 's). With this in mind, we define a process that maps the trimming proportion, α 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 Σ 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.