Maximal correlation coefficient

A measure of dependence of two random variables $ X $ and $ Y $, defined as the least upper bound of the values of the correlation coefficients between the real random variables $ \phi _ {1} ( X) $ and $ \phi _ {2} ( Y) $, which are functions of $ X $ and $ Y $ such that $ {\mathsf E} \phi _ {1} ( X) = {\mathsf E} \phi _ {2} ( Y) = 0 $ and $ {\mathsf D} \phi _ {1} ( X) = {\mathsf D} \phi _ {2} ( Y) = 1 $:

$$ \rho ^ {*} ( X , Y ) = \ \sup {\mathsf E} [ \phi _ {1} ( X) \phi _ {2} ( Y) ] . $$

If this least upper bound is attained at $ \phi _ {1} = \phi _ {1} ^ {*} ( X) $ and $ \phi _ {2} = \phi _ {2} ^ {*} ( Y) $, then the maximal correlation coefficient between $ X $ and $ Y $ is equal to the correlation coefficient of $ \phi _ {1} ^ {*} ( X) $ and $ \phi _ {2} ^ {*} ( Y) $. The maximal correlation coefficient has the property: $ \rho ^ {*} ( X , Y ) = 0 $ is necessary and sufficient for the independence of $ X $ and $ Y $. If there is a linear correlation between the variables, then the maximal correlation coefficient coincides with the usual correlation coefficient.

References

[1]	O.V. Sarmanov, "The maximum correlation coefficient (symmetric case)" Dokl. Akad. Nauk SSSR , 120 : 4 (1958) pp. 715–718 (In Russian)
[2]	O.V. Sarmanov, Dokl. Akad. Nauk SSSR , 53 : 9 (1946) pp. 781–784
[3]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)

Comments

See also Canonical correlation.

References

[a1]	H. Gebelein, "Das statistische Problem der Korrelation als Variations- und Eigenwertproblem und sein Zusammenhang mit der Ausgleichungrechnung" Z. Angew. Math. Mech. , 21 (1941) pp. 364–379
[a2]	R. Koyak, "On measuring internal dependence in a set of random variables" Ann. Statist. , 15 (1987) pp. 1215–1229