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.

翻译：对于二元概率分布，支撑集上某一点附近的局部依赖性通常被定义为概率密度函数对数的二阶导数。然而，该定义缺乏在边缘分布变换下的不变性，而这通常是依赖性度量所需满足的一个准则。在本研究中，我们针对Copula考察了\textit{相对局部依赖性}——我们将其定义为局部依赖性与概率密度函数之比。通过使用这一概念，我们指出典型的Copula可以被刻画为相应偏微分方程的解，特别强调了Frank copula的相对局部依赖性保持恒定。我们使用模拟数据展示了相对局部依赖性的估计与可视化。此外，我们提出了一类局部依赖性与概率密度函数的$k$次幂成正比的Copula，并作为一个示例，我们展示了从两个代表性Copula——Frank copula和Farlie-Gumbel-Morgenstern (FGM) copula——的密度函数中推导出的一个新发现的关系。