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Symmetric Gini Covariance and Correlation

Standard Gini covariance and Gini correlation play important roles in measuring the dependence of random variables with heavy tails. However, the asymmetry brings a substantial difficulty in interpretation. In this paper, we propose a symmetric Gini-type covariance and a symmetric Gini correlation ($ρ_g$) based on the joint rank function. The proposed correlation $ρ_g$ is more robust than the Pearson correlation but less robust than the Kendall's $τ$ correlation. We establish the relationship between $ρ_g$ and the linear correlation $ρ$ for a class of random vectors in the family of elliptical distributions, which allows us to estimate $ρ$ based on estimation of $ρ_g$. The asymptotic normality of the resulting estimators of $ρ$ are studied through two approaches: one from influence function and the other from U-statistics and the delta method. We compare asymptotic efficiencies of linear correlation estimators based on the symmetric Gini, regular Gini, Pearson and Kendall's $τ$ under various distributions. In addition to reasonably balancing between robustness and efficiency, the proposed measure $ρ_g$ demonstrates superior finite sample performance, which makes it attractive in applications.