Unbiased Estimation of Central Moments by using U-statistics

We obtain an estimator of the r th central moment of a distribution, which is unbiased for all distributions for which the first r moments exist. We do this by finding the kernel which allows the r th central moment to be written as a regular statistical functional. The U-statistic associated with this kernel is the unique symmetric unbiased estimator of the r th central moment, and, for each distribution, it has minimum variance among all estimators which are unbiased for all these distributions.