In many hypothesis testing problems such as one-sample and two-sample test problems, the test statistics are degenerate U-statistics. One of the challenges in practice is the computation of U-statistics for a large sample size. Besides, for degenerate U-statistics, the limiting distribution is a mixture of weighted chi-squares, involving the eigenvalues of the kernel of the U-statistics. As a result, it’s not straightforward to construct the rejection region based on this asymptotic distribution. In this research, we aim to reduce the computation complexity of degenerate U-statistics and propose an easy-to-calibrate test statistic by using the divide-and-conquer method. Specifically, we randomly partition the full n data points into kn even disjoint groups, and compute U-statistics on each group and combine them by averaging to get a statistic Tn. We proved that the statistic Tn has the standard normal distribution as the limiting distribution. In this way, the running time is reduced from O(n^m) to O( n^m/km_n), where m is the order of the one sample U-statistics. Besides, for a given significance level , it’s easy to construct the rejection region. We apply our method to the goodness of fit test and two-sample test. The simulation and real data analysis show that the proposed test can achieve high power and fast running time for both one and two-sample tests.
| Identifer | oai:union.ndltd.org:ndsu.edu/oai:library.ndsu.edu:10365/31777 |
| Date | January 2020 |
| Creators | Atta-Asiamah, Ernest |
| Publisher | North Dakota State University |
| Source Sets | North Dakota State University |
| Detected Language | English |
| Type | text/dissertation |
| Format | application/pdf |
| Rights | NDSU policy 190.6.2, https://www.ndsu.edu/fileadmin/policy/190.pdf |
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