Study on statistics associated with estimable parameters

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As estimators of an estimable parameter, U-statistics and V-statistics are well-known and have been discussed. We get the new statistic which are described by a linear combination of U-statistics and include these statistics. By introducing a new weight function we can get a new statistic. Our results are as follows: (1) By a new weight, we get a new S-statistic. For a non-degenerate kernel, V-statistic and S-statistic have no difference in the mean of second order efficiency. So we compare these two statistics by 4-th order efficiency. (2) As the sample size increases, the behavior of the linear combination of U-statistics can be described by a stochastic process on the interval [0,1]. We show the invariance principle that this process approaches to Brown motion. (3) The rate of convergence of the linear combination of U-statistics are evaluated with respect to the almost sure convergence. (4) We get the Edgeworth expansion of the linear combination of U-statistics. The above results are obtained for non-degenerate kernel. For degenerate kernel, the asymptotic distribution is complicated. Our results for degenerate kernel are as follows: (5) We get the asymptotic distribution of the linear combination of U-statistics. This result yields the new evaluation of the asymptotic distribution of the V-statistic, which is different from the previous fact. (6) We show the invariance principle of the linear combination of U-statistics. (7) We evaluate the rate of convergence that the linear combination of U-statistic approaches to the asymptotic distribution for degenerate kernel.