Sankhya: The Indian Journal of Statistics
1994, Volume 56, Series A, Pt. 2 , pp. 179--194
UNIFORM STRONG CONSISTENCY RATES FOR CONDITIONAL U-STATISTICS
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.
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