A Nonparametric Test for Trends in the Occurrence of Rare Events

Abstract

A nonparametric test for trends in the occurrence of rare events, based on the average position that the events occupy in the series, is presented. This test is formally identical to the Wilcoxon?Mann?Whitney test for the difference of means between two samples. In the present application, however, the range of values of the length N of the series for which accurate critical values are available has to be expanded considerably. Exact formulas for the p value of the test on the average position for number of events m = 2, 3, and 4 are given, as well as a recursive relation for general m. Since this procedure cannot in practical times be carried out beyond a small m, a combinatorial technique that allows the p-value computation with great accuracy is explained. For m greater than around 20 it is shown that the convergence to the normal distribution is good. The power of this nonparametric test is shown to be superior to that of tests based on the interevent times. The test is applied to a series of annual minimum temperatures and to a series of seasonal precipitation totals, thereby illustrating the practical advantages of this approach.