Abstract
Abstract Standard Gini correlation plays an important role in measuring the dependence between random variables with heavy-tailed distributions. It is based on the covariance between one variable and the rank of the other. Hence for each pair of random variables, there are two Gini correlations and they are not equal in general, which brings a substantial difficulty in interpretation. Recently, Sang et al (2016) proposed a symmetric Gini correlation based on the joint spatial rank function with a computation cost of O(n 2) where n is the sample size. In this paper, we study two symmetric and computationally efficient Gini correlations with the computational complexity of O(n log n). The properties of the new symmetric Gini correlations are explored. The influence function approach is utilized to study the robustness and the asymptotic behavior of these correlations. The asymptotic relative efficiencies are considered to compare several popular correlations under symmetric distributions with different tail-heaviness as well as an asymmetric log-normal distribution. Simulation and real data application are conducted to demonstrate the desirable performance of the two new symmetric Gini correlations.
Highlights
Measuring the strength of association and correlation between two random variables is of essential importance in many research elds
We have systematically studied two symmetric Gini correlations r(g ) and r(g ), which are the arithmetic and geometric means of the traditional Gini correlations γ and γ
We studied basic properties of r(g ) and r(g ), as well as their relationships to the correlation parameter in the elliptical distributions and log-normal distribution
Summary
Measuring the strength of association and correlation between two random variables is of essential importance in many research elds. For any continuous bivariate distribution H with nite rst moment, the in uence functions of the traditional Gini correlations are given by IF((u, v)T; γ , H) = γ Ρ arcsin(ρ/ ) + ( − ρ ))/ − ρ , which is smaller than vγ , the asymptotic variance of γ This means that the symmetric Gini correlation is more statistically e cient than the standard Gini correlation under normal distributions. The Fisher consistent correlation coe cients estimate the same parameter and their asymptotic variances and statistical e ciencies are comparable. In this case, the Pearson correlation has extremely large asymptotic variances, the result agreeing well with [19, 23]. ASV of Kendall’s tau is the most e cient in this case
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