Title: Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables Show Chinese title

Title: 两个相关伽马变量的归一化差异比的分布和矩

Abstract: We consider two random variables $X$ and $Y$ following correlated Gamma distributions, characterized by identical scale and shape parameters and a linear correlation coefficient $\rho$. Our focus is on the parameter: \[ D(X,Y) = \frac{|X - Y|}{X + Y}, \] which appears in applied contexts such as dynamic speckle imaging, where it is known as the \textit{Fujii index}. In this work, we derive a closed-form expression for the probability density function of $D(X,Y)$ as well as analytical formulas for its moments of order $k$. Our derivation starts by representing $X$ and $Y$ as two correlated exponential random variables, obtained from the squared magnitudes of circular complex Gaussian variables. By considering the sum of $k$ independent exponential variables, we then derive the joint density of $(X,Y)$ when $X$ and $Y$ are two correlated Gamma variables. Through appropriate varable transformations, we obtain the theoretical distribution of $D(X,Y)$ and evaluate its moments analytically. These theoretical findings are validated through numerical simulations, with particular attention to two specific cases: zero correlation and unit shape parameter. Show Chinese abstract

Abstract: 我们考虑两个服从相关Gamma分布的随机变量$X$和$Y$，其具有相同的尺度和形状参数以及线性相关系数$\rho$。 我们的关注点是参数：\[ D(X,Y) = \frac{|X - Y|}{X + Y}, \]，它在动态散斑成像等应用情境中出现，此时它被称为\textit{藤井指数}。 在本工作中，我们推导了$D(X,Y)$的概率密度函数的闭式表达式，以及其阶数为$k$的矩的解析公式。 我们的推导从将$X$和$Y$表示为两个相关的指数随机变量开始，这些变量来自于圆复高斯变量的模平方。 通过考虑$k$个独立指数变量的和，我们随后推导出当$X$和$Y$为两个相关的伽马变量时，$(X,Y)$的联合密度。 通过适当的变量变换，我们得到了$D(X,Y)$的理论分布，并对其矩进行了分析计算。 这些理论结果通过数值模拟得到验证，特别关注了两种具体情况：零相关性和单位形状参数。