From the Huang–Kotz FGM distribution to Baker’s bivariate distribution

Huang and Kotz (1999) [17] considered a modification of the Farlie–Gumbel–Morgenstern (FGM) distribution, introducing additional parameters, and paved the way for many research papers on modifications of FGM distributions allowing high correlation. The first part of the present paper is a review of recent developments on bivariate Huang–Kotz FGM distributions and their extensions. In the second part a class of new bivariate distributions based on Baker’s system of bivariate distributions is considered. It is shown that for a model of a given order, this class of distributions with fixed marginals which are based on pairs of order statistics constructed from the bivariate sample observations of dependent random variables allows higher correlation than Baker’s system. It also follows that under certain conditions determined by Lin and Huang (2010) [21], the correlation for these systems converges to the maximum Fréchet–Hoeffding upper bound as the sample size tends to infinity.