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October, 1994 On Some Asymptotic Properties of $U$ Statistics and One-Sided Estimates
Arup Bose, Ratan Dasgupta
Ann. Probab. 22(4): 1715-1724 (October, 1994). DOI: 10.1214/aop/1176988479


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.


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Arup Bose. Ratan Dasgupta. "On Some Asymptotic Properties of $U$ Statistics and One-Sided Estimates." Ann. Probab. 22 (4) 1715 - 1724, October, 1994.


Published: October, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0836.60027
MathSciNet: MR1331200
Digital Object Identifier: 10.1214/aop/1176988479

Primary: 60F15
Secondary: 60G42 , 62G05 , 62G20 , 62G30

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

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 4 • October, 1994
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