Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909534760468480 |
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| author | Colin, Elise Ossikovski, Razvigor |
| author_facet | Colin, Elise Ossikovski, Razvigor |
| contents | We consider two random variables $X$ and $Y$ following correlated Gamma distributions, characterized by identical scale and shape parameters and a linear correlation coefficient $ρ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables Colin, Elise Ossikovski, Razvigor Statistics Theory Mathematical Physics Instrumentation and Detectors Optics We consider two random variables $X$ and $Y$ following correlated Gamma distributions, characterized by identical scale and shape parameters and a linear correlation coefficient $ρ$. 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. |
| title | Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables |
| topic | Statistics Theory Mathematical Physics Instrumentation and Detectors Optics |
| url | https://arxiv.org/abs/2503.08808 |