Relationships between two extensions of Farlie-Gumbel-Morgenstern distribution

Summary

In order to increase the dependence between two random variablesX andY obeying the type of Farlie-Gumbel-Morgenstern (FGM) distribution, Johnson and Kotz (1977,Commun. Statist.,6, 485–496) introduced the (k−1)-iterationFGM distribution:

$$H_{1k} = FG + \sum\limits_{j = 1}^k {\alpha _{1j} (FG)^{[j/2] + 1} (\bar F\bar G)^{[(j + 1)/2]} } $$

whereF andG are the respective marginal distributions ofX andY. Recently, Huang and Kotz (1984,Biometrika,71, 633–636) found the natural parameter space ofH ₁₂ for arbitraryabsolutely continuous distributionsF andG. We extend their result to arbitrarycontinuous distributionsF andG and propose another (k−1)-iterationFGM distribution:

$$H_{2k} = FG + \sum\limits_{j = 1}^k {\alpha _{2j} (FG)^{[(j + 1)/2]} (\bar F\bar G)^{[j/2] + 1} } $$

For someF andG the correlation coefficient forH_2k is greater than that forH_1k.

Further, we find the conditions onF andG under whichH_1k andH_2k have the same natural parameter space. We also find that for arbitrary symmetric distributionsF andG with finite means, the covariances betweenX andY are the same whatever the joint distributionH_ik (i=1, 2) they have. A result of Schucany, Parr and Boyer (1978,Biometrika,65, 650–653) about the correlation coefficient forFGM distribution is extended to arbitrary distributionsF andG. The multivariate case is also discussed.