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Monotone Dependence

George Kimeldorf and Allan R. Sampson
The Annals of Statistics
Vol. 6, No. 4 (Jul., 1978), pp. 895-903
Stable URL: http://www.jstor.org/stable/2958865
Page Count: 9
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Monotone Dependence
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Abstract

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 $\sup \{\operatorname{corr} \lbrack f(X), g(Y)\rbrack\},$ where the $\sup$ is taken over all pairs of suitable monotone functions f and g.

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