Geodesic
On the maximum correlation coefficient
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 1, pp. 191-197 Cet article a éte moissonné depuis la source Math-Net.Ru

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For an arbitrary random vector $(X,Y)$ and an independent random variable $Z$ it is shown that the maximum correlation coefficient between $X$ and $Y+\lambda Z$ as a function of $\lambda$ is lower semicontinuous everywhere and continuous at zero where it attains its maximum. If, moreover, $Z$ is in the class of self-decomposable random variables, then the maximal correlation coefficient is right continuous, nonincreasing for $\lambda\geqslant 0$ and left continuous, nondecreasing for $\lambda \leqslant 0$. Independent random variables $X$ and $Z$ are Gaussian if and only if the maximum correlation coefficient between $X$ and $X+\lambda Z$ equals the linear correlation between them. The maximum correlation coefficient between the sum of $n$ arbitrary independent identically distributed random variables and the sum of the first $m of these equals $\sqrt{m/n}$ (previously proved only for random variables with finite second moments, where it amounts also to the linear correlation). Examples provided reveal counterintuitive behavior of the maximum correlation coefficient for more general $Z$ and in the limit $\lambda \to \infty$.
Keywords: dependence, maximum correlation
Mots-clés : self-decomposable random variables. 
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W. Bryc; A. Dembo; A. Kagan. On the maximum correlation coefficient. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 1, pp. 191-197. http://geodesic.mathdoc.fr/item/TVP_2004_49_1_a13/

[1] Breiman L., Friedman J. H., “Estimating optimal transformations for multiple regression and correlation. [With discussion and with a reply by the authors.]”, J. Amer. Statist. Assoc., 80:391 (1985), 580–619 | DOI | MR | Zbl

[2] Bryc W., Smoleński W., “On the stability problem for conditional expectation”, Statist. Probab. Lett., 15:1 (1992), 41–66 | DOI | MR

[3] Csáki P., Fisher J., “On the general notion of maximal correlation”, Magyar Tud. Akad. Mat. Kutató Int. Közl., 8 (1963), 27–51 | MR | Zbl

[4] Dembo A., Kagan A., Shepp L. A., “Remarks on the maximum correlation coefficient”, Bernoulli, 7:2 (2001), 343–350 | DOI | MR | Zbl

[5] Gebelein H., “Das statistische Problem der Korrelation als Variations- und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung”, Z. Angew. Math. Mech., 21 (1941), 364–379 | DOI | MR

[6] Hirschfeld H. O., “A connection between correlation and contingency”, Proc. Cambridge Philos. Soc., 31 (1935), 520–524 | DOI

[7] Lancaster H. O., “Some properties of the bivariate normal distribution considered in the form of a contingency table”, Biometrika, 44 (1957), 289–292 | Zbl

[8] Lukach E., Kharakteristicheskie funktsii, Nauka, M., 1979, 424 pp. | MR

[9] Rényi A., “On measures of dependence”, Acta Math. Acad. Sci. Hungar., 10 (1959), 441–451 | DOI | MR | Zbl

[10] Rosenblatt M., Markov Processes. Structure and Asymptotic Behavior, Springer-Verlag, New York, Heidelberg, 1971, 268 pp. | MR

[11] Sarmanov O. V., “Maksimalnyi koeffitsient korrelyatsii (simmetrichnyi sluchai)”, Dokl. AN SSSR, 120:4 (1958), 715–718 | MR | Zbl

[12] Sarmanov O. V., “Maksimalnyi koeffitsient korrelyatsii (nesimmetrichnyi sluchai)”, Dokl. AN SSSR, 121:1 (1958), 52–55 | MR | Zbl

[13] Witsenhausen H. S., “On sequences of pairs of dependent random variables”, SIAM J. Appl. Math., 28 (1975), 100–113 | DOI | MR | Zbl