Monotonic Dependence Functions of Bivariate Distributions

Abstract

A new characterization of monotonic dependence is given here proceeding in a natural way from the consideration of a type of dependence weaker than quadrant dependence. More precisely, each bivariate distribution of $(X, Y)$ is transformed onto a pair of functions $^\mu{X, Y}$ and $^\mu{Y, X}$ defined on the interval $0 < p < 1$ and taking values from [-1, 1], with $^\mu{X, Y}(p)$ being a suitably normalized expected value of $X$ under the condition that $Y$ exceeds its $p$th quantile. The usefulness of these functions as a kind of measures of the strength of monotonic dependence as well as their close relation to regression functions is demonstrated. It is also suggested that these functions and their sample analogues could serve as useful tools in modelling and solving some statistical decision problems.