# Improved mathematical models of statistical regularities in precipitation

## TL;DR

This paper develops improved mathematical models for statistical regularities in precipitation, providing better asymptotic approximations for wet period characteristics and introducing new methods to define extremal precipitation events.

## Contribution

It introduces novel asymptotic models for precipitation characteristics using limit theorems and compares new extremal precipitation definitions with existing methods.

## Key findings

- Asymptotic distribution of wet period duration matches empirical data.
- Maximum daily precipitation follows a tempered gamma mixture distribution.
- Total precipitation volume approximates a generalized gamma distribution.

## Abstract

The paper presents improved mathematical models and methods for statistical regularities in the behavior of some important characteristics of precipitation: duration of a wet period, maximum daily and total precipitation volumes within a such period. The asymptotic approximations are deduced using limit theorems for statistics constructed from samples with random sizes having the generalized negative binomial (GNB) distribution. It demonstrates excellent concordance with the empirical distribution of the duration of wet periods measured in days. The asymptotic distribution of the maximum daily precipitation volume within a wet period turns out to be a tempered scale mixture of the gamma distribution with the scale factor having the Weibull distribution, whereas the asymptotic approximation to the total precipitation volume for a wet period turns out to be the generalized gamma (GG) distribution. Two approaches to the definition of abnormally extremal precipitation are presented. The first approach is based on an excess of a certain quantile of the asymptotic distribution of the maximum daily precipitation. The second approach is based on the GG model for the total precipitation volume. The corresponding statistical test is compared with a previously proposed one based on tha classical gamma distribution using real precipitation data.