Almost sure representations of weighted -statistics with applications

Winfried Stute

Journal of Nonparametric Statistics, 2008, vol. 20, issue 3, 191-205

Abstract: Let X1, …, Xn be a sample of i.i.d. random variables on the real line. In this paper, we introduce and study weighted U-statistics of the type Sn=(n(n−1))−1 ∑i¬=j J(i/n+1, j/n+1) h(Xi:n, Xj:n), where J and h are given bivariate functions and X1:n≤···≤Xn:n are the pertaining order statistics of the sample. Under weak integrability conditions on Jh, we derive an almost sure representation of Sn in terms of a U-statistic and a remainder which is roughly of the order n−1. Specification of J and h yields representations for L-statistics and various robust measures of spread. The method also applies in a two-sample as well as in a censored data situation.

Date: 2008
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DOI: 10.1080/10485250801999438

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