Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 2 , pp. 179--194

UNIFORM STRONG CONSISTENCY RATES FOR CONDITIONAL U-STATISTICS

By

ARUSHARKA SEN, Indian Statistical Institute

SUMMARY. Let (Xi, Yi)ni=1 be a random sample from a bivariate distribution. We want to estimate m(t): E(h(Y1,…, Yk) | X1=t1, …, Xk=tk) where k £ n, t=(t1, …, tk) Î Rk, and h: Rk ® R is s.t. E|h| < ¥ . Stute (1991) has proposed a conditional U-statistic Un(t). As an estimator for m(t). Here we establish rates of uniform strong convergence of Un(t) to m(t) under suitable assumptions. We make use of a ‘randomly weighted ‘ empirical process of U-statistic structure on Rk. Our principle tool is a sharp probability inequality proved be Alexander (1984).

AMS (1990) subject classification. Primary 62G07; secondary 62G20, 60F15.

Key words and phrases. Conditional $U$-statistics, uniform strong consistency, empirical processes of $U$-statistic structure, Vapnik-Cervonenkis theory, $V-C$ graph class, deviation measurability.

Full Paper (PDF)

This article in mathematical reviews.