A new measure of association between random variables (Q1683636)

A measure of association between two continuous random variables \(X\) and \(Y\), \(\alpha(X,Y)\), is proposed. It is based on the covariance between \(X\) and the log-odds rate associated to \(Y\). Thus, it measures the association between \(X\) and a random variable with standard logistic distribution. \(\alpha(X,Y)\) lies in the range \([-1,1]\) and the extremes of the range, i.e., \(-1\) and \(1\), are attainable by the Fréchet bivariate minimal and maximal distributions, respectively. These maximum and minimum values are achieved if \(Y\) is a monotone, respectively increasing or decreasing function of \(X\). If \(X\) and \(Y\) have normal distribution, \(\alpha(X,Y)\) equals their Pearson correlation coefficient. For the joint distribution of \(X\) and \(Y\) in the Farlie-Gumbel-Morgenstern (FGM) family of distributions, an explicit form of \(\alpha(X,Y)\) is derived. In the FGM family it depends only on the distribution of \(X\) and does not depend on that of \(Y\). For non-negative random variables, \(\alpha(X,Y)\) is represented in terms of the mean residual and mean inactivity functions. For independent random variables \(X\) and \(Y\), \(\alpha(X,Y)=\alpha(Y, X)=0\). \(\alpha(X,Y)=-\alpha(-X,Y)=\alpha(X,-Y)=\alpha(-X,-Y)\). \(\alpha(X,Y)\) is invariant for changes in the location and scale of \(X\) and \(Y\). If the joint distribution of \(X\) and \(Y\) is exchangeable, then \(\alpha(X,Y)=\alpha(Y,X)\).