STUDY OF NONPARAMETRIC STATISTICAL INFERENCE

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As estimators of estimable parameters, we consider three statistics which are U-statistic, V-statistic and limit of Bayes estimate. This limit of Bayes estimate, which we say LB-statistic, is obtained from Bayes estimate of estimable parameter based on Dirichlet process by letting its parameter tend to zero. For the estimable parameter with non-degenerate kernel, the asymptotic relative efficiencies of LB-statistic with respect to U-statistic and V-statistic, and that of V-statistic with respect to U-statistic are equal to one. To see difference between these three statistics, we evaluate deficiencies among LB-statistic, U-statistic and V-statistic.With respect to the expression of LB-statistics with degenerate kernel as a linear combination of U-statistics, we show the properties of its components which are unbiasedness and degeneracy. Using these we give its asymptotic distribution. By the same methods, we show the asymptotic distribution of V-statistic with degenerate kernel. For the V-statistic, its asymptotic distribution is given by Borovskikh (1996 ), p.113, whose form and method are different from us. We confirm two are equivalent.Associated with an estimable parameter, we consider a new class of statistics. It is described by a linear combination of U-statistics and includes V-statistic and LB-statistic as special cases. The kernel is assumed to be non-degenerate. For the difference between this linear combination and the well-known U-statistic, we derive upper bounds of its absolute moments about the origin and the mean. Using these upper bounds we show Berry-Esseen bound for the linear combination of U-statistics with exact expressions.