Distributed inference for degenerate U-statistics

In many hypothesis testing problems, the test statistics are degenerate U-statistics. One of the challenges in practice is the computation of U-statistics for large dataset. In this paper, we aim to reduce the computation complexity of degenerate U-statistics by using the divide-and-conquer method. Specifically, we partition the full n data points into k(n) even disjoint groups, compute the U-statistics on each group, and combine them by averaging. In this way, the running time is reduced to O(nmknm-1), where m is the order of the U-statistics. Besides, we study the optimal test rate of the divide-and-conquer methods. For degenerate U-statistics, the optimal rate is n , the same as the optimal test rate for the nondegenerate case. The simulation and real data confirm that the proposed method has high power and faster running time.