Article

Remarks on the Maximum Correlation Coefficient

Department of Mathematics, Stanford University, Palo Alto, California, United States
Bernoulli (Impact Factor: 1.16). 06/2000; 7(2). DOI: 10.2307/3318742
Source: OAI

ABSTRACT

The maximum correlation coefficient between partial sums of independent and identically distributed random variables with finite second moment equals the classical (Pearson) correlation coefficient between the sums, and thus does not depend on the distribution of the random variables. This result is proved, and relations between the linearity of regression of each of two random variables on the other and the maximum correlation coefficient are discussed.

    • "For random variables that take only a finite number of values, Sethuraman [18] gave a procedure to estimate the MC from the sample, and gave the asymptotic distribution of this estimate under the null hypothesis of independence. Dembo et al. [19] and Novak [20] studied the MC between partial sums of independent and identically distributed random variables. More recently, Yenigün et al. [21] considered the computation of MC in contingency tables and proposed an independence test. "
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    • "If, we restrict ϕ 1 , ϕ 2 to linear function, then MCC is the classical Pearson correlation coefficient, For further discussion on maximum correlation coefficient we refer to [2], [11]. "
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    • "Another exception is provided by the surprising result of Dembo et al. (2001), and its subsequent extensions given by Bryc et al. (2005) and Yu (2008). In its general form the result states that for any independent identically distributed (i.i.d.) nondegenerate r.v.'s X 1 , "
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    ABSTRACT: We provide a method that enables the simple calculation of the maximal correlation coefficient of a bivariate distribution, under suitable conditions. In particular, the method readily applies to known results on order statistics and records. As an application we provide a new characterization of the exponential distribution: Under a splitting model on independent identically distributed observations, it is the (unique, up to a location-scale transformation) parent distribution that maximizes the correlation coefficient between the records among two different branches of the splitting sequence.
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