Topics in U-statistics and Risk Estimation

We consider estimating the variance of a general U-statistic when it is used as an unbiased estimator of the parameter of interesttheta=E(K) where K is the kernel function. Long established results demonstrate the asymptotic normality of U-statistics and their asymptotic variance under regularity conditions. However,these asymptotic results are based on the assumption that the sample size n goes to infinity; they are not so reliable when n is notlarge or the kernel size m is not negligible compared with n. In addition, it can be seen that the asymptotic variance is always optimistic. Onthe other hand, the exact finite sample variance of a U-statistic is complicated in form when m is large. We consider an alternative approach to estimate its variance which has a relatively simple form. This variance estimator is the best unbiased estimator and therefore is applicable even for the cases that m/n is a fixed fraction. We also consider methods to estimate the U-statistic and its variance by "m out of n" resampling. Especially, two resampling schemes have been developed, both of which provide an unbiased realization of the unbiased variance estimator.In order to further investigate the proposed method, we apply it in risk estimation under the context of nonparametric density estimation. We constructed U-statistic form risk estimators based on L2 loss and Kullback-Leibler loss respectively, the former of which is comparable to the bagged CV score introduced in Hall and Robinson (2009). To evaluate the proposed variance estimator in risk estimation, we have carried out a simulation comparison with some bootstrap variance estimators.