k-U统计量的渐进正态性

Asymptotic Normality of k-U Statistic

摘要: 称 U_n^m, k=\sum _ 1 \leq i_1 < \cdots < i_m \leq k g\left(X_i_1, \cdots, X_i_m\right)+\sum_i_m-k+1 \leq i_1 <\cdots < i_m-1 < i_m \sum_k+1 \leq i_m \leq n g\left(X_i_1, \cdots, X_i_m\right) 为 k-U 统计量, 其中 g\left(x_1, \cdots, x_m\right) 是一对称函数, k 为小于等于 n 的自然数, k 可能依赖于 n. 这一表达形式是一类统计量, 在 k=n 时, \left( ^n\right)^-1 U_n^m, n 就是 U - 统计量. 本文证明了 U_n^m, k 的渐进正态性.

Abstract: U_n^m, k=\sum_1 \leq i_1<\cdots<i_m \leq k g\left(X_i_1, \cdots, X_i_m\right)+\sum_i_m-k+1 \leq i_1<\cdots<i_m-1<i_m \sum_k+1 \leq i_m \leq n g\left(X_i_1, \cdots, X_i_m\right) is called the k-U statistic, where g\left(x_1, \cdots, x_m\right) is a symmetric function, k is a positive integer, and equal to or less than n and probably dependent on n. This statistic is a class of statistic and when k=n,\binomnm^-1 U_n^m, n is U-statistic. The asymptotic normality of the k-U statistic will be studied in this paper.