Multivariate Classes of GB2 Distributions with Applications

José María Sarabia,Montserrat Guillén,Faustino Prieto,Vanesa Jordá

doi:10.3390/math9010072

Abstract

The general beta of the second kind distribution (GB2) is a flexible distribution which includes several relevant parametric families of distributions. This distribution has important applications in earnings and income distributions, finance and insurance. In this paper, several multivariate classes of the GB2 distribution are proposed. The different multivariate versions are based on two simple univariate representations of the GB2 distribution. The first type of multivariate distributions are constructed from a stochastic dependent representations defined in terms of gamma random variables. Using this representation and beginning by two particular multivariate GB2 distributions, multivariate Singh–Maddala and Dagum income distributions are presented and several properties are obtained. Then, a general multivariate GB2 distribution is introduced. The second type of multivariate distributions are based on a generalization of the distribution of the order statistics, which gives place to multivariate GB2 distribution with support above the diagonal. We discuss the role of these families in modeling bivariate income distributions. Finally, an empirical application is given, where we show that a multivariate GB2 distribution can be useful for modeling compound precipitation and wind events in the whole range.

Highlights

The use of parametric functional forms for the study of earnings and income distributions has been well documented in the literature
It can be noted that all the parameter estimates are statistically significant at a 0.05 level of significance, assuming the asymptotic normality of the maximum likelihood estimates
The first class is based on stochastic dependent representations defined in terms of gamma random variables

Summary

Introduction

The use of parametric functional forms for the study of earnings and income distributions has been well documented in the literature (see [1,2]). Among the existing parametric models, we emphasize the general beta of the second kind distribution (GB2). The GB2 is a flexible and wide distribution which includes many well-known models as special or limiting cases. This family provides an excellent description of income distributions with a few parameters. An economic origin of this distribution is available: Parker’s model of optimizing firm behavior characterizes an earnings distribution of the GB2 type [3]. The GB2 distribution has been used in several fields of economics and business, including modeling of income and wealth data (e.g., [4,5,6,7,8]), Lorenz ordering [9,10], unemployment duration data [11], regression models with non-negative random variables [12], actuarial losses [13] and option pricing [14]

Methods

Results

Conclusion