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

====Comments====
See also [[Canonical correlation|Canonical correlation]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Koyak,   "On measuring internal dependence in a set of random variables"  ''Ann. Statist.'' , '''15'''  (1987)  pp. 1215–1229</TD></TR></table>