Relative local dependence of bivariate copulas

Abstract: For a bivariate probability distribution, local dependence around a single point on the support is often formulated as the second derivative of the logarithm of the probability density function. However, this definition lacks the invariance under marginal distribution transformations, which is often required as a criterion for dependence measures. In this study, we examine the \textit{relative local dependence}, which we define as the ratio of the local dependence to the probability density function, for copulas. By using this notion, we point out that typical copulas can be characterised as the solutions to the corresponding partial differential equations, particularly highlighting that the relative local dependence of the Frank copula remains constant. The estimation and visualization of the relative local dependence are demonstrated using simulation data. Furthermore, we propose a class of copulas where local dependence is proportional to the $k$-th power of the probability density function, and as an example, we demonstrate a newly discovered relationship derived from the density functions of two representative copulas, the Frank copula and the Farlie-Gumbel-Morgenstern (FGM) copula.