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Two‐stage U‐statistics for Hypothesis Testing

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  • HWAI‐CHUNG HO
  • GRACE S. SHIEH

Abstract

. A U‐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 U‐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 U‐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 U‐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 U‐statistic in almost all cases under two settings.

Suggested Citation

  • Hwai‐Chung Ho & Grace S. Shieh, 2006. "Two‐stage U‐statistics for Hypothesis Testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 861-873, December.
    • Handle: RePEc:bla:scjsta:v:33:y:2006:i:4:p:861-873
    DOI: 10.1111/j.1467-9469.2006.00516.x
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