JSM 2005 - Toronto

Abstract #303040

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Legend: = Applied Session, = Theme Session, = Presenter
Activity Number: 103
Type: Contributed
Date/Time: Monday, August 8, 2005 : 8:30 AM to 10:20 AM
Sponsor: IMS
Abstract - #303040
Title: Multivariate L-Moments
Author(s): Peng Xiao*+ and Robert Serfling
Companies: The University of Texas Southwestern Medical Center at Dallas and The University of Texas Southwestern Medical Center at Dallas
Address: Mathematical Sciences Department, Richardson, TX, 75083, United States
Keywords: L-moments ; multivariate ; nonparametric descriptive measures ; U-statistics
Abstract:

Univariate L-moments (Hosking, 1990) exist for all orders and even determine the distribution, provided only that the mean exists, which serves as the first L-moment. The L-moments parallel the usual central moments in role and interpretation but are defined more broadly, being more applicable, for example, to heavy-tailed data. For order greater than one, extension to the multivariate case has remained a conceptually challenging open problem. Here, we provide a solution in which matrix-valued multivariate versions of L-moments are formulated for arbitrary order greater than one. The first multivariate L-moment is just the mean vector; the second is a Gini covariance matrix; the third a "coskewness matrix;" the fourth a "cokurtosis matrix;" the fifth a "cobimodality matrix," etc. These all require only first moment assumptions on the component variables, and can be used in a style similar to the classical covariance matrix. Special features, properties, and examples will be discussed, as well as estimation of the multivariate L-moments using matrix-valued U-statistic estimators.

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