Statistical Tests for Extreme Precipitation Volumes

Abstract

The approaches, based on the negative binomial model for the distribution of duration of the wet periods measured in days, are proposed to the definition of extreme precipitation. 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 as well as the total precipitation volume over a wet period. The first approach to the definition (and determination) of 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 extreme, if it exceeds 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 extremely large can be formulated as the homogeneity hypothesis of a sample from the gamma distribution. Two equivalent tests are proposed for testing this hypothesis. Both of these tests deal with the relative contribution of the total precipitation volume for a wet period to the considered set (sample) of successive wet periods. Within the second approach it is possible to introduce the notions of relatively and absolutely extreme 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 extreme large precipitation volume increases.

Probability Models and Statistical Tests for Extreme Precipitation Based on Generalized Negative Binomial Distributions

“…In order to have a flexible model of a mixing distribution that is "responsible" for the description of statistical regularities of the manifestation of external stochastic factors, we suggest to use the GG distributions defined by the density (2). The class of GG distributions was first described as a unitary family in 1962 by E. Stacy [11] as the class of probability distributions simultaneously containing both Weibull and gamma distributions.…”

“…In this paper, we continue the research we started in [1,2]. We develop the mathematical models for statistical regularities in precipitation proposed in the papers mentioned above.…”

“…The theoretical grounds for this model are provided by the law of large numbers for random sums in which the number of summands has the GNB distribution. It turns out that, as compared to the ordinary negative binomial (NB) model (see [2]), the additional exponent power parameter in the corresponding GG distribution matches slow global climate trends. Hence, the hypothesis that the total precipitation volume during a certain wet period is abnormally large can be re-formulated as the homogeneity hypothesis of a sample from the GG distribution.…”

“…Based on these models, two approaches are proposed to the definition of abnormally extremal precipitation. These approaches can be regarded as a further development of those proposed in [2].…”

“…In [2], we proposed a rather reasonable approach to the unambiguous (algorithmic) determination of extreme or abnormally heavy total precipitation for a wet period. This approach was based on the NB model for the duration of wet periods measured in days, and, as a consequence, on the distribution of the total precipitation volume during a wet period.…”