A simple method for obtaining the maximal correlation coefficient and related characterizations

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.

[1]  Yaming Yu.  On the maximal correlation coefficient , 2008 .

[2]  K. C. Cherian.  A Bi-Variate Correlated Gamma-Type Distribution Function , 1941 .

[3]  N. Papadatos,et al.  An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds , 2011 .

[4]  Angelo Efoévi Koudou,et al.  Lancaster bivariate probability distributions with Poisson, negative binomial and gamma margins , 1998 .

[5]  A. Dembo,et al.  On the Maximum Correlation Coefficient , 2005 .

[6]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[7]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation. , 1985 .

[8]  H. Gebelein.  Das statistische Problem der Korrelation als Variations‐ und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung , 1941 .

[9]  H. O. Lancaster.  Some properties of the bivariate normal distribution considered in the form of a contingency table , 1957 .

[10]  Narayanaswamy Balakrishnan,et al.  Bounds on expectation of order statistics from a finite population , 2003 .

[11]  G. Terrell,et al.  A Characterization of Rectangular Distributions , 1983 .

[12]  N. Balakrishnan,et al.  Continuous Bivariate Distributions , 2009 .

[13]  Amir Dembo,et al.  Remarks on the maximum correlation coefficient , 2001 .

[14]  Fernando López-Blázquez,et al.  Upper and lower bounds for the correlation ratio of order statistics from a sample without replacement , 2006 .

[15]  R. Jain,et al.  Records , 1973, Tempo.

[16]  M. Manser,et al.  Chi-Squared Distribution , 2010 .

[17]  Tamás F. Móri,et al.  An extremal property of rectangular distributions , 1985 .