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:
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:
For someF andG the correlation coefficient forH2k is greater than that forH1k.
Further, we find the conditions onF andG under whichH1k andH2k 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 distributionHik (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.
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References
Cambanis, S. (1977). Some properties and generalizations of multivariate Eyraud-Gumbel-Morgenstern distributions,J. Multivar. Anal.,7, 551–559.
Huang, J. S. and Kotz, S. (1984). Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions,Biometrika,71, 633–636.
Johnson, N. L. and Kotz, S. (1975). On some generalized Farlie-Gumbel-Morgenstern distributions,Commun. Statist.,4, 415–427.
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.
Royden, H. L. (1972).Real analysis (2nd ed.), the Macmillan Company, New York.
Schucany, W., Parr, W. C. and Boyer, J. E. (1978). Correlation structure in Farlie-Gumbel-Morgenstern distributions,Biometrika,65, 3, 650–653.
Shaked, M. (1975). A note on the exchangeable generalized Farlie-Gumbel-Morgenstern distributions,Commun. Statist.,4, 711–721.
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Work finally completed while visiting Department of Statistics, Stanford University. The author was partially supported by the Chinese National Science Council.
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Lin, G.D. Relationships between two extensions of Farlie-Gumbel-Morgenstern distribution. Ann Inst Stat Math 39, 129–140 (1987). https://doi.org/10.1007/BF02491454
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DOI: https://doi.org/10.1007/BF02491454