Non-Parametric Inference for Gini Covariance and its Variants
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Non-Parametric Inference for Gini Covariance and its Variants Sudheesh K. Kattumannil Indian Statistical Institute, Chennai, India
N. Sreelakshmi Indian Institute of Technology, Chennai, India
N. Balakrishnan McMaster University, Hamilton, Canada Abstract We obtain simple non-parametric estimators of Gini-based covariance, correlation and regression coefficients. We then establish the consistency and asymptotic normality of the proposed estimators. We provide an explicit formula for finding the asymptotic variance of the estimators. We also discuss jackknifed versions of the proposed estimators for reducing the bias of the estimators in case of small sample sizes. Finally, we evaluate the finitesample performance of these estimators through on Monte Carlo simulations from a bivariate Pareto distribution. AMS (2000) subject classification. Primary: 62G05, Secondary: 62G20. Keywords and phrases. Gini mean difference, Gini covariance, Gini correlation, Gini index, Gini regression coefficient.
1 Introduction The measurement of economic inequality in a particular economy is a major topic of study in economic and statistical literatures. In 1912, Corrado Gini proposed a measure of variability, known as Gini mean difference (GMD), as the expected absolute difference between two randomly drawn observations from the same population. GMD has been generalized to several inequality measures by putting different weights on the expectation. Gini index is one of the prominent measures derived from GMD, which has also been generalized into a family of inequality measures, known as Single Series Gini (S-Gini) family of indices, that depends on a single parameter (Yitzhaki, 1983). For more details on GMD and other measures derived from
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it, interested readers may refer to Barrett and Donald (2009) and Yitzhaki and Schechtman (2013). The representation of GMD in terms of covariance facilitates the concepts of Gini covariance, Gini correlation and Gini regression coefficients. Schechtman and Yitzhaki (1987) proposed the concept of Gini covariance, which has been used widely to measure the dependence of heavy-tailed distributions. The Gini covariance has an advantage while analyzing bivariate data as it is defined based on both variate values and ranks of the values. For example, in insurance and financial modelling, loss and profit and loss distributions are generally skewed and heavy-tailed, with non-linear dependence structures among variables. So, researchers associate one variable with ranks of another variable. Motivated by this, Furman and Zitikis (2017) discussed the use of Gini correlation in the classical Capital Asset Pricing Model (CAPM) into insurance and finance without relying on the Gaussian assumption. The role of Gini covariance and related quantities as coherent risk measures was discussed by Furman et al. (2017), who also discussed the conditional version of Gini covariance and related quantities. Some other works in this direction are due to Gribkova and Zitikis (2017) and Gribkova and Zitikis (2019)
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