A Unified Approach to Construct Correlation Coefficient Between Random Variables

TL;DR

This paper introduces a unified correlation measure for continuous random variables based on covariance and distribution functions, encompassing existing measures and extending the Pearson correlation with broader applicability.

Contribution

It proposes a new unified correlation measure derived from covariance and distribution functions, generalizing existing correlation measures and including a variant of Pearson's coefficient.

Findings

The proposed measure ranges between -1 and 1 under mild conditions.

It subsumes some existing correlation measures.

Numerical examples demonstrate its applicability to well-known bivariate distributions.

Abstract

Measuring the correlation (association) between two random variables is one of the important goals in statistical applications. In the literature, the covariance between two random variables is a widely used criterion in measuring the linear association between two random variables. In this paper, first we propose a covariance based unified measure of variability for a continuous random variable X and we show that several measures of variability and uncertainty, such as variance, Gini mean difference, cumulative residual entropy, etc., can be considered 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, based on the covariance between X and H^{-1}G(Y) (known as the Q-transformation of H on G) where H is a continuous DF. 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 the suggested index ranges between [-1,1] where the extremes of the range, i.e., -1 and 1, are attainable by the Frechet bivariate minimal and maximal DFs, respectively. A special case of the proposed correlation measure leads to a variant of Pearson correlation coefficient which, as a measure of strength and direction of the linear relationship between X and Y, has absolute values greater than or equal to the Pearson correlation. The results are examined numerically for some well known bivariate DFs.