A <b>"U"</b>-statistic is not easy to apply or cannot be applied in hypothesis testing when it is degenerate or has an indeterminate degeneracy under the null hypothesis. A class of two-stage <b>"U"</b>-statistics (TU-statistics) is proposed to remedy these drawbacks. Both the asymptotic distributions under the null and the alternative of TU-statistics are shown to have simple forms. When the degeneracy is indeterminate, the Pitman asymptotic relative efficiency of a TU-statistic dominates that of the incomplete <b>"U"</b>-statistics. If the kernel is degenerate under the null hypothesis but non-degenerate under the alternative, a TU-statistic is proved to be more powerful than its corresponding <b>"U"</b>-statistic. Applications to testing independence of paired angles in ecology and marine biology are given. Finally, a simulation study shows that a TU-statistic is more powerful than its corresponding incomplete <b>"U"</b>-statistic in almost all cases under two settings. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..
