*One-sample U-statistics can be regarded as generalizations of means. They are sums of dependent variables, but we show them to be asymptotically normal by the projection method. Certain interesting test statistics, such as the Wilcoxon statistics and Kendall’s τ-statistic, are one-sample U-statistics. The Wilcoxon statistic for testing a difference in location between two samples is an example of a two-sample U-statistic. The Cramér–von Mises statistic is an example of a degenerate U-statistic*.

Let *X*_{1},..., *X*_{n} be a random sample from an unknown distribution. Given a known function *h*, consider estimation of the “parameter”

In order to simplify the formulas, it is assumed throughout this section ...

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