J. Japan Statist. Soc., Vol. 31 (No. 1), pp. 85-98, 2001

Asymptotic Comparisons of U-Statistics, V-Statistics and Limits of Bayes Estimates by Deficiencies

Toshifumi Nomachi and Hajime Yamato

Abstract. 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, called LB-statistic in this paper, 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. We show asymptotic differences among LB-statistic, U-statistic and V-statistic by using the deficiency.

Key words and phrases: Dirichlet process, Efficiency, Estimable parameter, Kernel, Degree.

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