Maximizing the Correlation of Grouped Observations

Abstract

A continuous bivariate distribution is grouped by intervals on the margins so as to obtain a discrete bivariate distribution. Marginal variates are considered which are the scores (or mean values) on the intervals of the first canonical variables associated with the ungrouped distribution. A method is suggested for choosing the end points of the intervals so as to obtain the maximum possible correlation between these marginal variates. The solution is exact when only one margin is grouped and approximate when both are. Tables of interval end points (in terms of the c.d.f.) and of the resulting correlation are given for the case of a bivariate normal distribution.