Gini covariance matrix and its affine equivariant version

Dang, Xin; Sang, Hailin; Weatherall, Lauren

doi:10.1007/s00362-016-0842-z

Cited by 8 publications

(8 citation statements)

References 42 publications

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“…In addition we define

{cov}_{g} false(X, X false) : = 2 𝔼 X R_{1} false(Z false) = 𝔼 \frac{{false(X_{1} - X_{2} false)}^{2}}{∥ {boldZ}_{1} - {boldZ}_{2} ∥};

{cov}_{g} false(Y, Y false) : = 2 𝔼 Y R_{2} false(Z false) = 𝔼 \frac{{false(Y_{1} - Y_{2} false)}^{2}}{∥ {boldZ}_{1} - {boldZ}_{2} ∥} .

We see that not only the Gini covariance between X and Y but also Gini variances of X and of Y are defined jointly through the spatial rank. Dang, Sang, & Weatherall () considered the Gini covariance matrix

{boldΣ}_{g} = 2 𝔼 Z {boldr}^{⊤} false(Z false)

. The covariances defined in , , and are elements of

{boldΣ}_{g}

for two‐dimensional random vectors.…”

Section: Symmetric Gini Covariance and Correlationmentioning

confidence: 99%

“…Although implies Theorem 4, it is hard to check regularity conditions for the von Mises expansion . We instead prove it in the Appendix using the multivariate delta method and the asymptotic normality of the sample Gini covariance matrix, which is based on the U ‐statistics theory (Dang, Sang, & Weatherall, ).…”

Section: Estimationmentioning

confidence: 99%

“…Hence

{boldΣ}^{1 / 2} r u + μ

is a sample from a bivariate Kotz

false(μ, Σ false)

distribution. For additional details, see Dang, Sang, & Weatherall ().…”

Section: Estimationmentioning

confidence: 99%

“…We now provide an affine invariant version of

ρ_{g}

, denoted as

ρ_{G}

, in order to gain the invariance property under heterogeneous changes. This is based on the affine equivariant (AE) Gini covariance matrix

{boldΣ}_{G}

proposed by Dang, Sang, & Weatherall ().…”

Section: The Affine Invariant Version Of Symmetric Gini Correlationmentioning

confidence: 99%

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Symmetric Gini covariance and correlation

2016

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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.

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“…In addition we define

{cov}_{g} false(X, X false) : = 2 𝔼 X R_{1} false(Z false) = 𝔼 \frac{{false(X_{1} - X_{2} false)}^{2}}{∥ {boldZ}_{1} - {boldZ}_{2} ∥};

{cov}_{g} false(Y, Y false) : = 2 𝔼 Y R_{2} false(Z false) = 𝔼 \frac{{false(Y_{1} - Y_{2} false)}^{2}}{∥ {boldZ}_{1} - {boldZ}_{2} ∥} .

{boldΣ}_{g} = 2 𝔼 Z {boldr}^{⊤} false(Z false)

. The covariances defined in , , and are elements of

{boldΣ}_{g}

for two‐dimensional random vectors.…”

Section: Symmetric Gini Covariance and Correlationmentioning

confidence: 99%

Section: Estimationmentioning

confidence: 99%

“…Hence

{boldΣ}^{1 / 2} r u + μ

is a sample from a bivariate Kotz

false(μ, Σ false)

distribution. For additional details, see Dang, Sang, & Weatherall ().…”

Section: Estimationmentioning

confidence: 99%

“…We now provide an affine invariant version of

ρ_{g}

, denoted as

ρ_{G}

, in order to gain the invariance property under heterogeneous changes. This is based on the affine equivariant (AE) Gini covariance matrix

{boldΣ}_{G}

proposed by Dang, Sang, & Weatherall ().…”

Section: The Affine Invariant Version Of Symmetric Gini Correlationmentioning

confidence: 99%

See 2 more Smart Citations

Symmetric Gini covariance and correlation

2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

Two symmetric and computationally efficient Gini correlations

Vanderford

Sang

Dang

2020

Dependence Modeling

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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(n2) 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.

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Bounds for Gini’s mean difference based on first four moments, with some applications

Yin

2022

Stat Papers

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Gini covariance matrix and its affine equivariant version

Cited by 8 publications

References 42 publications

Symmetric Gini covariance and correlation

Symmetric Gini covariance and correlation

Two symmetric and computationally efficient Gini correlations

Bounds for Gini’s mean difference based on first four moments, with some applications

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