Lin, Gwo Dong Relationships between two extensions of Farlie-Gumbel-Morgenstern distribution. (English) Zbl 0633.62016 Ann. Inst. Stat. Math. 39, 129-140 (1987). 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 Cited in 12 Documents MSC: 62E10 Characterization and structure theory of statistical distributions 62E15 Exact distribution theory in statistics 62H10 Multivariate distribution of statistics Keywords:correlation coefficient; continuity; Farlie-Gumbel-Morgenstern (FGM) distribution; absolutely continuous distributions; arbitrary continuous distributions; natural parameter space; multivariabe case Citations:Zbl 0382.62040; Zbl 0555.62050 PDFBibTeX XMLCite \textit{G. D. Lin}, Ann. Inst. Stat. Math. 39, No. 1--2, 129--140 (1987; Zbl 0633.62016) Full Text: DOI References: [1] Cambanis, S. (1977). Some properties and generalizations of multivariate Eyraud-Gumbel-Morgenstern distributions,J. Multivar. Anal.,7, 551-559. · Zbl 0377.62017 · doi:10.1016/0047-259X(77)90066-5 [2] Huang, J. S. and Kotz, S. (1984). Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions,Biometrika,71, 633-636. · Zbl 0555.62050 [3] Johnson, N. L. and Kotz, S. (1975). On some generalized Farlie-Gumbel-Morgenstern distributions,Commun. Statist.,4, 415-427. · Zbl 0342.62006 · doi:10.1080/03610927508827258 [4] Johnson, N. L. and Kotz, S. (1977). On some generalized Farlie-Gumbel-Morgenstern distributions—II: Regression, correlation and further generalizations,Commun. Statist.,6, 485-496. · Zbl 0382.62040 · doi:10.1080/03610927708827509 [5] Royden, H. L. (1972).Real analysis (2nd ed.), the Macmillan Company, New York. · Zbl 0704.26006 [6] Schucany, W., Parr, W. C. and Boyer, J. E. (1978). Correlation structure in Farlie-Gumbel-Morgenstern distributions,Biometrika,65, 3, 650-653. · Zbl 0397.62033 · doi:10.1093/biomet/65.3.650 [7] Shaked, M. (1975). A note on the exchangeable generalized Farlie-Gumbel-Morgenstern distributions,Commun. Statist.,4, 711-721. · Zbl 0315.62010 · doi:10.1080/03610927508827281 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.