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{1176988479,
author = {Bose, Arup and Dasgupta, Ratan},
title = {On Some Asymptotic Properties of $U$ Statistics and One-Sided Estimates},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1715-1724},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988479}
}
Bose, Arup; Dasgupta, Ratan. On Some Asymptotic Properties of $U$ Statistics and One-Sided Estimates. Ann. Probab., Tome 22 (1994) no. 4, pp. 1715-1724. http://gdmltest.u-ga.fr/item/1176988479/