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Almost sure representations of weighted -statistics with applications

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  • Winfried Stute

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.

Suggested Citation

  • Winfried Stute, 2008. "Almost sure representations of weighted -statistics with applications," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(3), pages 191-205.
    • Handle: RePEc:taf:gnstxx:v:20:y:2008:i:3:p:191-205
    DOI: 10.1080/10485250801999438
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    References listed on IDEAS

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    1. A. Ehsanes Saleh, 1976. "Hodges-lehmann estimate of the location parameter in censored samples," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 28(1), pages 235-247, December.
    2. Cheng, Philip E. & Lin, Gwo Dong, 1987. "Maximum likelihood estimation of a survival function under the koziol-green proportional hazards model," Statistics & Probability Letters, Elsevier, vol. 5(1), pages 75-80, January.
    3. Stute, W., 1994. "Lp-Convergence of Conditional U-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 71-82, October.
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