Abstract:
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Gini covariance as a measure of dependence between two variables plays an important role in Gini methodology. In this paper, we propose a new symmetric Gini-type covariance based on a joint rank function. As a result, a new correlation called the symmetrical Gini correlation (Gcor, $\rho_g$) is also defined. We study the properties of Gcor. Its influence function is derived and shows that it is more robust than Pearson correlation but less robust than Kendall $\tau$. The relation between the symmetric Gini correlation $\rho_g$ and the linear correlation $\rho$ is established for a class of random vectors in the family of elliptical distributions. With this relationship, an estimator of $\rho$ is obtained from estimating $\rho_g$. We study the asymptotic normality of this estimator through two approaches: one from influence function and the other from U-statistics and Delta method. We compare asymptotic efficiencies of linear correlation estimators based on Gini, Pearson and Kendall $\tau$ under various distributions.The proposed one not only balances between robustness and efficiency, its superior finite sample behavior also makes it advantageous in application.
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