A discrete analogue of Terrell's characterization of rectangular distributions

TL;DR

This paper extends Terrell's 1983 characterization of rectangular distributions, showing that the maximum correlation coefficient for a pair from a discrete distribution occurs only for discrete rectangular (uniform) distributions.

Contribution

It proves that the maximal correlation coefficient in the discrete case is achieved only by discrete rectangular distributions, paralleling the continuous case.

Findings

Maximum correlation coefficient is attained only for discrete rectangular distributions.

The result parallels Terrell's continuous case characterization.

Uses Hahn polynomials in the proof.

Abstract

George R. Terrell (1983, {Ann. Probab., vol. 11(3), pp. 823--826) showed that the Pearson coefficient of correlation of an ordered pair from a random sample of size two is at most one-half, and the equality is attained only for rectangular (uniform over some interval) distributions. In the present note it is proved that the same is true for the discrete case, in the sense that the correlation coefficient attains its maximal value only for discrete rectangular (uniform over some finite lattice) distributions. MSC: Primary 60E15; 62E10; Secondary 62G30. Key words and phrases: discrete rectangular distribution; order statistics; Hahn polynomials; Pearson coefficient of correlation.