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

Victor Korolev,Andrey Gorshenin

doi:10.3390/math8040604

Abstract

Mathematical models are proposed for statistical regularities of maximum daily precipitation within a wet period and total precipitation volume per wet period. The proposed models are based on the generalized negative binomial (GNB) distribution of the duration of a wet period. The GNB distribution is a mixed Poisson distribution, the mixing distribution being generalized gamma (GG). The GNB distribution demonstrates excellent fit with real data of durations of wet periods measured in days. By means of limit theorems for statistics constructed from samples with random sizes having the GNB distribution, asymptotic approximations are proposed for the distributions of maximum daily precipitation volume within a wet period and total precipitation volume for a wet period. It is shown that the exponent power parameter in the mixing GG distribution matches slow global climate trends. The bounds for the accuracy of the proposed approximations are presented. Several tests for daily precipitation, total precipitation volume and precipitation intensities to be abnormally extremal are proposed and compared to the traditional PoT-method. The results of the application of this test to real data are presented.

Highlights

In this paper, we continue the research we started in [1,2]
The results of fitting the generalized negative binomial (GNB) distribution to real data are presented and demonstrate excellent concordance of the GNB model with the empirical distribution of the duration of wet periods measured in days
This approach was based on the negative binomial (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

Summary

Introduction

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. 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. The corresponding mathematical models are theoretically based on limit theorems of probability theorems that yield unambiguous asymptotic approximations, which are used as adequate mathematical models This approach gives an unambiguous algorithm for the determination of extreme or abnormally heavy total precipitation that does not involve statistical significance problems owing to the low occurrence of such (relatively rare) events. The tests appeared to be applicable to total precipitation volumes over wet periods and to the precipitation intensities (the ratios of total precipitation volumes per wet periods to the durations of the corresponding wet periods measured in days)

Generalized Negative Binomial Model for the Duration of Wet Periods

Statistical Tests for Anomalously Extreme Total Precipitation Volumes

Conclusions and Discussion