Generalized Radius Processes for Elliptically Contoured Distributions
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 re...
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Published in | Journal of the American Statistical Association Vol. 100; no. 471; pp. 1036 - 1045 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Alexandria, VA
Taylor & Francis
01.09.2005
American Statistical Association Taylor & Francis Ltd |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 0162-1459 1537-274X |
DOI: | 10.1198/016214504000002023 |