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 $k=n,\binom{n}{m}^{-1} U_n^{m, n}$ is U-statistic. The asymptotic normality of the k-U statistic will be studied in this paper.
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