Monotone Dependence
George Kimeldorf, Allan R. Sampson
The Annals of Statistics · 1978 · Volume 6 · Issue 4
Abstract
Random variables $X$ and $Y$ are mutually completely dependent if there exists a one-to-one function $g$ for which $P\lbrack Y = g(X)\rbrack = 1.$ 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 $\sup \{\operatorname{corr} \lbrack f(X), g(Y)\rbrack\},$ where the $\sup$ is taken over all pairs of suitable monotone functions $f$ and $g.$
Keywords
62H05, 62H20, association, Correlation, monotone correlation, monotone dependence, sup correlation
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