Bivariate Kumaraswamy Models via Modified FGM Copulas: Properties and Applications

Indranil Ghosh

doi:10.3390/jrfm10040019

Abstract

A copula is a useful tool for constructing bivariate and/or multivariate distributions. In this article, we consider a new modified class of FGM (Farlie–Gumbel–Morgenstern) bivariate copula for constructing several different bivariate Kumaraswamy type copulas and discuss their structural properties, including dependence structures. It is established that construction of bivariate distributions by this method allows for greater flexibility in the values of Spearman’s correlation coefficient, ρ and Kendall’s τ .

Highlights

Over the last decade or so, there has been a growing interest in constructing various bivariate distributions and study their dependence structure
One of the most important parametric family of copulas is the Farlie–Gumbel–Morgenstern (FGM, ) family defined as C(u, v) = uv[1 + θ(1 − u)(1 − v)], (u, v) ∈ (0, 1), where θ ∈ [−1, 1]. This family of copulas have the following properties. Such family is derived from so called Farlie–Gumbel–Morgenstern distributions considered by Morgenstern (1956) and Gumbel (1960) and further developed by Farlie (1960)
We consider the construction of bivariate KW distributions and discuss some of their structural properties

Summary

Introduction

Over the last decade or so, there has been a growing interest in constructing various bivariate distributions and study their dependence structure. With some specific choice of the functions A(x) = 1 − x, and B(y) = 1 − y (see Equation (1) of Bairamov et al (2001), they have shown that the admissible range for the association parameter is between [−1, 1], while the Pearson correlation coefficient ρ between X and Y will never exceed 1/3 This fuels working in this direction in the sense of considering a modified FGM class and using it as a pivot for constructing bivariate Kumaraswamy models. We will consider some specific choices of Φ(u) and Ψ(v) to construct bivariate Kumaraswamy type copulas

Bivariate KW-FGM Type Models

Simulation from a Bivariate Copula

Application in Risk Management

An Application to Insurance Data

Findings

Conclusions