Statistical tests for daily and total precipitation volumes to be abnormally extremal
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In this paper, two approaches are proposed to the definition of abnormally extremal precipitation. These approaches are based on the negative binomial model for the distribution of duration of wet periods measured in days. This model demonstrates excellent fit with real data and provides a theoretical base for the determination of asymptotic approximations to the distributions of the maximum daily precipitation volume within a wet period and of the total precipitation volume over a wet period. The asymptotic distribution of the maximum daily precipitation volume within a wet period turns out to be a tempered Snedecor-Fisher distribution whereas the total precipitation volume for a wet period turns out to be the gamma distribution. These asymptotic approximations are deduced using limit theorems for statistics constructed from samples with random sizes. The first approach to the definition of abnormally extreme precipitation is based on the tempered Snedecor-Fisher distribution of the maximum daily precipitation. According to this approach, a daily precipitation volume is considered to be abnormally extremal, if it exceeeds a certain (pre-defined) quantile of the tempered Snedecor-Fisher distribution. The second approach is based on that the total precipitation volume for a wet period has the gamma distribution. Hence, the hypothesis that the total precipitation volume during a certain wet period is abnormally large can be formulated as the homogeneity hypothesis of a sample from the gamma distribution. Two equivalent tests are proposed for testing this hypothesis. Within the second approach it is possible to introduce the notions of relatively abnormal and absolutely abnormal precipitation volumes. The results of the application of these tests to real data are presented yielding the conclusion that the intensity of wet periods with abnormally large precipitation volume increases.
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