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

#### **Victor Korolev 1,2,3,4 and Andrey Gorshenin 1,2,3,\***


Received: 4 April 2020; Accepted: 14 April 2020; Published: 16 April 2020

**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.

**Keywords:** precipitation; limit theorems; statistical test; generalized negative binomial distribution; generalized gamma distribution; asymptotic approximations; extreme order statistics; random sample size

**MSC:** 60F05; 62G30; 62E20; 62P12; 65C20
