JSM 2005 - Toronto

Abstract #303029

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Activity Number: 103
Type: Contributed
Date/Time: Monday, August 8, 2005 : 8:30 AM to 10:20 AM
Sponsor: IMS
Abstract - #303029
Title: Multivariate Spatial U-Quantile Functions with Examples
Author(s): Weihua Zhou*+ 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 Dept, Richardson, TX, 75083,
Keywords: multivariate ; spatial quantiles ; U-statistics ; Bahadur representation ; nonparametric inference
Abstract:

For sample quantiles of univariate data, Bahadur (1966) and Kiefer (1967, 1970) gave an almost sure convergence rate for a suitable linear approximation of the estimation error. The Bahadur result was extended by Choudhury & Serfling (1988) and Arcones (1996) to "U-quantiles"---generalized quantile functions based on the evaluations of a real-valued kernel over an arbitrary dataset. Here, we consider multivariate versions of U-quantiles, employing vector-valued kernels and the multivariate spatial quantile notion introduced by Dudley and Koltchinskii (1992) and Chaudhuri (1996). We extend to spatial U-quantiles a Bahadur-Kiefer representation established by Koltchinskii (1994) for ordinary spatial quantiles. Interesting new multivariate nonparametric estimators arise through the multivariate spatial U-quantiles formulation and support related hypothesis tests based on corresponding centered rank functions. Applications to be discussed include multivariate Hodges-Lehmann location and spread statistics, nonparametric estimation of multiple regression slope coefficients, and a new dispersion matrix parameter with sample analogue estimator having breakdown point 0.293.

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