Tukey Depth-based Multivariate Trimmed Means

• Activity Number: 202
• Type: Topic Contributed
• Date/Time: Tuesday, August 13, 2002 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics*
• Author(s): Jean-Claude Masse*+
• Affiliation(s): Université Laval
• Keywords: central limit theorem ; Tukey depth ; multivariate trimmed mean ; Hadamard differentiability

Abstract

Depth functions provide a convenient way of measuring the centrality of a point with respect to a multivariate probability distribution. Informally speaking, points with high depth are viewed as being close to the \lq\lq center'' of the distribution and points with low depth are understood as belonging to the tails. Given a $d$-dimensional dataset, depth-based trimmed means can thus be defined by averaging those points whose depth with respect to the empirical distribution is higher than a certain level. This talk will focus on two types of Tukey depth-based multivariate trimmed means. Under two sets of conditions, several central limit theorems are obtained, all of them based on Hadamard differentiability of the location functionals and the delta method. In the univariate setting, a central limit theorem due to Stigler is derived as a special case.