# Maximal correlation coefficient

From Encyclopedia of Mathematics

A measure of dependence of two random variables and , defined as the least upper bound of the values of the correlation coefficients between the real random variables and , which are functions of and such that and :

If this least upper bound is attained at and , then the maximal correlation coefficient between and is equal to the correlation coefficient of and . The maximal correlation coefficient has the property: is necessary and sufficient for the independence of and . 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 |

**How to Cite This Entry:**

Maximal correlation coefficient. I.O. Sarmanov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Maximal_correlation_coefficient&oldid=19195

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098