Study on functional s of distributions and convex combinations of U-statistics

作为与分布泛函相关的统计量，冯·米塞斯(1947)的V-统计量和Hoeffding(1948)的U-统计量是众所周知的，也被许多研究者研究过。Yamato(1977)引入了一种新的统计量--LB-统计量。Toda和Yamato(2001)引入了U-统计量的一个线性组合(U-统计量的一个凸组合)，它包含了这三个统计量。我们研究了U-统计量的这种凸组合的性质。我们估计了U-统计量的凸组合的大偏差，这对于估计尾概率是有用的。在核退化的情况下，得到了U统计量凸组合的泛函极限定理(不变性原理)，并得到了它的Edgeworth展开式。首先按样本量的-1阶得到标准化统计量，然后按样本量的-1/2阶得到学生化统计量。我们还得到了样本容量为-1阶的学生化统计量的Edgeworth展开式。该统计量与两样本U-统计量具有相同的渐近正态分布。为了了解这两个统计量之间的差异，我们得到了样本容量为-1/2阶的两样本U-统计量凸组合的Edgeworth展开。

As statistics related with a functional of distribution, V-statistic of von Mises (1947) and U-statistic of Hoeffding (1948) are well-known and have been studied by many researchers. Yamato (1977) introduced a new statistic, LB-statistic. Toda and Yamato (2001) introduced a linear combination of U-statistics (a convex combination of U-statistic), which include these three statistics.We have studied the properties of this convex combination of U-statistics.We evaluated large deviations of the convex combination of U-statistics, which is useful to evaluate the tail probability. In case where the kernel is degenerate, we obtained a functional limit theorem (invariance principle) of a convex combination of U-statistics.We also obtained its Edgeworth expansion. At first we got it for the standardized statistic by the order-1 of the sample size and furthermore got it for the studentized statistic by the order -1/2 of the sample size. We also got the Edgeworth expansion for the studentized statistic by the order-1 of the sample size.Next, we introduced a convex combination of two-sample U-statistics. This statistic has the asymptotic normality as same as the two-sample U-statistic. To see the difference between these two statistics, we got the Edgeworth expansion of convex combination of two-sample U-statistics with the order -1/2 of the sample size.