Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 3 , pp. 476-479

A CHARACTERIZATION OF FARLIE-GUMBEL-MORGENSTERN DISTRIBUTIONS VIA SPEARMAN'S RHO AND CHI-SQUARE DIVERGENCE

By

ROGER B. NELSEN, Lewis and Clark College

SUMMARY. We show that among all absolutely continuous bivariate distributions with fixed marginals and a given value $\rho _0$ of Spearman's rho, where $|\rho _0| \le 1/3$, the one whose joint density is closest (in the sense of $\chi ^2$-divergence) to the density of independent random variables is the Farlie-Gumbel-Morgenstern distribution with parameter 3$\rho_0$. This result provides alternate interpretations of Spearman's rho for Farlie-Gumbel-Morgenstern distributions.

AMS (1980) subject classification. 62E10; 62H05.

Key words and phrases. Farlie-Gumbel-Morgenstern distributions, Spearman's rho, chi-square divergence.

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