On the distribution of the sum of dependent standard normally distributed random variables using copulas

TL;DR

This paper investigates how different copulas affect the distribution of the sum of two dependent standard normal variables, providing analytical formulas and numerical evaluations to highlight significant differences in the resulting distributions.

Contribution

It derives closed-form expressions for the joint density using various copulas and evaluates their impact on the sum's distribution through numerical methods.

Findings

Significant differences in the sum distribution depending on the copula used.

Higher quantiles show deviations over 10% between copulas.

Analytical formulas enable precise modeling of dependencies.

Abstract

The distribution function of the sum $Z$ of two standard normally distributed random variables $X$ and $Y$ is computed with the concept of copulas to model the dependency between $X$ and $Y$. By using implicit copulas such as the Gauss- or t-copula as well as Archimedean Copulas such as the Clayton-, Gumbel- or Frank-copula, a wide variety of different dependencies can be covered. For each of these copulas an analytical closed form expression for the corresponding joint probability density function $f_{X,Y}$ is derived. We apply a numerical approximation algorithm in Matlab to evaluate the resulting double integral for the cumulative distribution function $F_Z$. Our results demonstrate, that there are significant differencies amongst the various copulas concerning $F_Z$. This is particularly true for the higher quantiles (e.g. $0.95, 0.99$), where deviations of more than $10\%$ have been noticed.