A Modified Hybrid Gamma and Generalized Pareto Distribution for Precipitation Data

Abstract

This study introduces a modified hybrid gamma and generalized Pareto distribution. Prior to this, we define a general spliced distribution and its corresponding gamma distribution, which is part of the head, and a generalized Pareto (GP) distribution, which is part of the tail. We then examine the threshold conditions for the modified hybrid gamma and GP distribution and defined probability density function. Also, we derive the negative log-likelihood function of the modified hybrid gamma and GP distribution and estimate approximate maximum likelihood estimates using the differential evolution algorithm for each simulation to minimize it. Moreover, by presenting the mean square error for each sample size, the model is evaluated according to the size of the sample. Finally, we use daily observed summer precipitation for Seoul, Korea, from 1961 to 2011, which includes 4692 data sets. We use 2051 data sets corresponding to wet conditions. As a result, the estimated threshold of the modified hybrid gamma and GP distribution is 0.1455. After deriving Fisher information through the Hessian matrix, we also present the standard error of the maximum likelihood estimator.