A Family of Correlation Coefficients Based on Extended Gini

The extended Gini is a family of measures of variability which is mainly used in the areas of finance and income distribution. Each index in the family is defined by specifying one parameter. The higher the parameter, the more weight is attached to the lower portion of the cumulative distribution. In this paper we list and investigate the properties of the equivalents of the correlation coefficient that are associated with the extended Gini family. In addition, we show that the extended Gini of a linear combination of random variables can be decomposed, in a way which includes the terms that are equivalent to the terms involved in the decomposition of the variance, plus additional terms that reflect additional properties of the random variables. The implication of these properties is that any decomposition that is performed with the coefficient of variation can be replicated by an infinite number of indices that are based on the Extended Gini coefficient.