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|Paper IPM / M / 16028||
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, based on the covariance between X and H−1G(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 that the suggested index ranges in [−1,1] where the extremes of the range, i.e., -1 and 1, are attainable by the Fr\acuteechet 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.
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