Two‐stage U‐statistics for Hypothesis Testing

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.

[1]  P. Halmos.  The Theory of Unbiased Estimation , 1946 .

[2]  W. Hoeffding.  A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[3]  T. W. Anderson,et al.  Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes , 1952 .

[4]  G. S. Watson,et al.  Goodness-of-fit tests on a circle. II , 1961 .

[5]  G. Gregory,et al.  Large Sample Theory for $U$-Statistics and Tests of Fit , 1977 .

[6]  H. Rubin,et al.  Asymptotic Distribution of Symmetric Statistics , 1980 .

[7]  Nicholas I. Fisher,et al.  Nonparametric measures of angular-angular association , 1981 .

[8]  P. Major.  Multiple Wiener-Ito Integrals: With Applications to Limit Theorems , 1981 .

[9]  N. Fisher,et al.  A correlation coefficient for circular data , 1983 .

[10]  S. Janson.  The asymptotic distributions of incomplete U-statistics , 1984 .

[11]  Nicholas I. Fisher,et al.  Correlation coefficients for random variables on a unit sphere or hypersphere , 1986 .

[12]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[13]  Stefun D. Leigh.  U-Statistics Theory and Practice , 1992 .

[14]  Modification of some goodness-of-fit statistics to yield asymptotically normal null distributions , 1993 .

[15]  Herman J. Bierens,et al.  Asymptotic Theory of Integrated Conditional Moment Tests , 1997 .

[16]  Qi Li,et al.  Central limit theorem for degenerate U-Statistics of Absolutely Regular Processes with Applications to Model Specification Testing , 1999 .

[17]  Ulric J. Lund.  Least circular distance regression for directional data , 1999 .

[18]  Harshinder Singh,et al.  Probabilistic model for two dependent circular variables , 2002 .

[19]  Grace S. Shieh,et al.  Inferences based on a bivariate distribution with von Mises marginals , 2005 .