Monotone Dependence

Random variables X and Y are mutually completely dependent if there exists a one-to-one function g for which P Y = g(X) = 1. An example is presented of a pair of random variables which are mutually completely dependent, but "almost" independent. This example motivates considering a new concept of dependence, called monotone dependence, in which g above is now required to be monotone. Finally, this monotone dependence concept leads to defining and studying the properties of a new numerical measure of statistical association between random variables X and Y defined by f(X), g(Y)\, where the is taken over all pairs of suitable monotone functions f and $g.$