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

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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.