Distribution of Quotient of Dependent and Independent Random Variables Using Copulas

Abstract

Determining distributions of the functions of random variables is a very important problem with wide applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X₁/X ₂ and the ratio of one variable over the sum of two variables Z = X₁/X₁+X₂ of two dependent or independent random variables X₁ and X₂ by using copulas to capture the structures between X₁ and X₂. We then extend the theory by establishing the density and distribution functions for the quotients Y = X₁/ X₂ and Z = X₁/X₁+X₂ of two dependent normal random variables X₁ and X₂ in case of Gaussian copulas. Thereafter, we develop the theory on the median for the ratios of both Y and Z on two normal random variables X₁ and X₂. Furthermore, the result of median for Z is also extended to a larger family of symmetric distributions and symmetric copulas of X₁ and X₂. Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall's coefficient. Our findings are useful for academics, practitioners, and policy makers.

Keywords: Copulas, Dependence structures, Quotient of random variables, Density functions, Distribution functions