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Relationships between two extensions of Farlie-Gumbel-Morgenstern distribution. (English) Zbl 0633.62016

N. L. Johnson and S. Kotz, Commun. Stat., Theory Methods A6, 485-496 (1977; Zbl 0382.62040) introduced the (k-1)-iteration Farlie- Gumbel-Morgenstern (FGM) distribution \[ H_{1k}=FG+\sum^{k}_{j=1}\alpha_{1j}(FG)^{[j/2]+1}(\bar F\bar G)^{[(j+1)/2]} \] where F and G are the marginal distributions. J. S. Huang and S. Kotz, Biometrika 71, 633-636 (1984; Zbl 0555.62050), found the natural parameter space of \(H_{12}\) for arbitrary absolutely continuous distributions F and G.
The present paper extends the latter result to arbitrary continuous distributions F and G and proposes another (k-1)-iteration FGM distribution: \[ H_{2k}=FG+\sum^{k}_{j=1}\alpha_{2j}(FG)^{[(j+1)/2]}(\bar F\bar G)^{[(j/2)+1]}. \] Further, the conditions are found on F and G under which \(H_{1k}\) and \(H_{2k}\) have the same natural parameter space. The multivariabe case and some other properties are also discussed.
Reviewer: J.Panaretos

MSC:

62E10 Characterization and structure theory of statistical distributions
62E15 Exact distribution theory in statistics
62H10 Multivariate distribution of statistics
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