Symmetric Gini covariance and correlation

Abstract

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.

Two symmetric and computationally efficient Gini correlations

“…where (X , Y ) T and (X , Y ) T are independently distributed from H. For Z = (X, Y) T from H with nite rst moment, the joint-rank based symmetric Gini correlation r (s) g [28] is de ned as…”

“…Some researchers [21,27] even list symmetry as one of the axioms of association measures. A symmetric Gini correlation was proposed in [4,28], which is based on the joint rank function. It is more statistically e cient than the standard Gini correlations, but it is not computationally e cient with O(n ) complexity, which means it is prohibitive for large n. Yitzhaki and Olkin [42] proposed two symmetric Gini correlations which are the arithmetic mean and geometric mean of the standard Gini correlations, respectively.…”

Jackknife empirical likelihood methods for Gini correlations and their equality testing

“…We do not include another popular correlation Kendall τ in the simulation. Its performance is referred to [16]. We generate 3000 samples of two different sample sizes (n = 20, 200) from two different bivariate distributions, namely, normal and t(5), with the scatter matrix Σ = 1 2ρ 2ρ 4 .…”

“…The average coverage probabilities and average lengths of confidence intervals as well as their standard deviations (in parenthesis) are presented in Tables 1, 2. Under elliptical distributions including normal and t distributions, the two Gini correlations and the Pearson correlation are equal to the linear correlation parameter ρ, that is, γ(X, Y ) = γ(Y, X) = ρ P = ρ ( [16]). Thus, all the listed methods in Table 1 and Table 2 are for the same quantity ρ.…”

“…(T, log W ) from N (0, Σ) with the scatter matrix Σ = 1 4ρ 4ρ 16 and consider δ 1 = γ(X, Y ) − γ(T, W ) and δ 2 = γ(Y, X) − γ(W, T ). As mentioned in Section 3, δ 1 := γ(X, Y ) − γ(T, W ) = 0 and δ 2 := γ(Y, X) − γ(W, T ) = 0.…”