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


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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    ABSTRACT: In most of the regression problems the first task is to select the most influential predictors explaining the response, and removing the others from the model. These problems are usually referred to as the variable selection problems in the statistical literature. Numerous methods have been proposed in this field, most of which address linear models. In this study we propose two variable selection criteria for regression based on two powerful dependence measures, maximal correlation and distance correlation. We focus on these two measures since they fully or partially satisfy the Rényi postulates for dependence measures, and thus they are able to detect nonlinear dependence structures. Therefore, our methods are considered to be appropriate in linear as well as nonlinear regression models. Both methods are easy to implement and they perform well. We illustrate the performances of the proposed methods via simulations, and compare them with two benchmark methods, stepwise Akaike information criterion and lasso. In several cases with linear dependence all four methods turned out to be comparable. In the presence of nonlinear or uncorrelated dependencies, we observed that our proposed methods may be favourable. An application of the proposed methods to a real financial data set is also provided.
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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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