A unified approach to constructing correlation coefficients between random variables

Abstract

Measuring the correlation between two random variables is an important goal in various statistical applications. The standardized covariance is a widely used criterion for measuring the linear association. In this paper, first, we propose a covariance-based unified measure of variability for a continuous random variable X and show that several measures of variability and uncertainty, such as variance, Gini mean difference and cumulative residual entropy arise as special cases. Then, we propose a unified measure of correlation between two continuous random variables X and Y, with distribution functions (DFs) F and G. Assuming that H is a continuous DF, the proposed measure is defined based on the covariance between X and the transformed random variable $H^{-1}G(Y)$ (known as the Q-transformation of H on G). We show that our proposed measure of association subsumes some of the existing measures of correlation. Under some mild condition on H, it is shown that the suggested index ranges in $[-1,1]$ where the extremes of the range, i.e., $-1$ and 1, are attainable by the Fréchet bivariate minimal and maximal DFs, respectively. A special case of the proposed correlation measure leads to a variant of the Pearson correlation coefficient which has absolute values greater than or equal to Pearson correlation. The results are examined numerically for some well known bivariate DFs.