The Annals of Probability

On Some Asymptotic Properties of $U$ Statistics and One-Sided Estimates

Arup Bose and Ratan Dasgupta

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Let $\{X_i, 1 \leq i \leq n\}$ be independent and identically distributed random variables. For a symmetric function $h$ of $m$ arguments, with $\theta = Eh(X_1,\ldots, X_m)$, we propose estimators $\theta_n$ of $\theta$ that have the property that $\theta_n \rightarrow \theta$ almost surely (a.s.) and $\theta_n \geq \theta$ a.s. for all large $n$. This extends the results of Gilat and Hill, who proved this result for $\theta = Eh(X_1)$. The proofs here are based on an almost sure representation that we establish for $U$ statistics. As a consequence of this representation, we obtain the Marcinkiewicz-Zygmund strong law of large numbers for $U$ statistics and for a special class of $L$ statistics.

Article information

Ann. Probab., Volume 22, Number 4 (1994), 1715-1724.

First available in Project Euclid: 19 April 2007

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Primary: 60F15: Strong theorems
Secondary: 60G42: Martingales with discrete parameter 62G05: Estimation 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

$U$ statistics $L$ statistics order statistics Marcinkiewicz-Zygmund strong law one-sided estimates


Bose, Arup; Dasgupta, Ratan. On Some Asymptotic Properties of $U$ Statistics and One-Sided Estimates. Ann. Probab. 22 (1994), no. 4, 1715--1724. doi:10.1214/aop/1176988479.

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