ASYMPTOTIC NORMALITY OF U-STATISTICS WHEN DIMEN-SIONS OF OBSERVATIONS ARE DIFFERENT AND SOME APPLICATIONS IN THE ANALYSIS OF FAMILIAL DATA

Abstract

Let X_i be n_i dimensional random variable for i=1, …, n. Hoeffding [2] proved the asymptotic normality of U-statistics when n_i≡k and n→∞ under some mild conditions. In this note, it is shown that the asymptotic normality still holds even if n_i_??_ k. Furthermore, some applications are given for estimators useful in the analysis of familial data.