A Simple Method for Obtaining the Maximal Correlation Coefficient and Related Characterizations

TL;DR

This paper introduces a straightforward method for calculating the maximal correlation coefficient in bivariate distributions, with applications to order statistics, records, and a new characterization of the exponential distribution.

Contribution

It presents a simple calculation method for maximal correlation and applies it to characterize the exponential distribution uniquely in a splitting model.

Findings

Provides a new characterization of the exponential distribution.

Offers a practical method for calculating maximal correlation.

Connects maximal correlation with order statistics and records.

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.