A New Hybrid Family of Probability Distributions for Fitting Heavy-tailed Data with Application to Finance

Abstract

The classical continuous univariate probability distributions, which contain one or two parameters, have been observed to break down when complexities exist in the structure of a data set such as when outliers are present, alongside observations centered around the mean. When a data set exhibits heterogeneity or exists in a multi-component form and it becomes impossible to use a single probability distribution to capture the distinct components of the data set, using a composite distribution to model the data set becomes plausible. This situation has led to the formulation of various hybrid or composite models where each component of the hybrid model handles the specific part of the data set that it is well suited for. Furthermore, the approach or method used in the formulation of these hybrid models plays a vital role in determining how meaningful the results obtained from them are. Several approaches or methods for formulating hybrid distributions have appeared in the literature, each with their own pros and cons. We present in this paper a general two-component hybrid model for fitting heterogeneous heavy-tailed data sets with tails to the right. The functional form of the two-component hybrid family is specified by the probability density function (pdf), cumulative distribution function (cdf) and the quantile function. Three members of the family using three different distributions for the right tail are presented.  A formal method based on maximum likelihood for the estimation of the parameters of the models belonging to the family is also presented. A Monte Carlo simulation study is carried out to determine the efficiency of the estimation method. An application to a real data set in finance is performed.