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Two‐stage U ‐statistics for Hypothesis Testing
Author(s) -
HO HWAICHUNG,
SHIEH GRACE S.
Publication year - 2006
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2006.00516.x
Subject(s) - mathematics , statistics , statistic , test statistic , statistical hypothesis testing , sufficient statistic , degeneracy (biology) , null hypothesis , u statistic , alternative hypothesis , degenerate energy levels , f test , econometrics , efficiency , biology , physics , bioinformatics , quantum mechanics , estimator
. 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.