×

A new bivariate binomial distribution. (English) Zbl 1056.62063

Summary: To bring correlations between binomial random variables is an important statistical problem with a lot of theoretical and practical applications. We provide a new formulation of bivariate binomial distributions in the sense that marginally each of the two random variables has a binomial distribution and they have some non-zero correlation in the joint distribution. A \(2\times 2\) contingency table is the immediate application of the proposed model.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
62H17 Contingency tables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aitken, A. C.; Gonin, H. T., Fourfold sampling with and without replacement, Proc. Roy. Soc. Edinburgh, 55, 114-125 (1935) · Zbl 0013.21405
[2] Consul, P. C., A simple urn-model dependent on pre-determined strategy, Sankhya Ser. B, 36, 391-399 (1974) · Zbl 0387.60017
[3] Crowder, M.; Sweeting, T., Bayesian inference for a bivariate binomial distribution, Biometrika, 76, 599-603 (1989) · Zbl 0674.62021
[4] Fisher, R. A., Statistical Methods for Research Workers (1950), Hafner Publishing Co: Hafner Publishing Co New York · JFM 60.1162.01
[5] Hamdan, M. A., Canonical expansion of the bivariate binomial distribution with unequal marginal indices, Internat. Statist. Rev., 40, 277-288 (1972) · Zbl 0244.62012
[6] Hamdan, M. A., A note on the trinomial distribution, Internat. Statist. Rev., 43, 219-220 (1975) · Zbl 0328.62009
[7] Hamdan, M. A.; Jensen, D. R., A bivariate binomial distribution and some applications, Austral. J. Statist., 18, 163-169 (1976) · Zbl 0398.62010
[8] Hamdan, M. A.; Martinson, E. O., Maximum likelihood estimation in the bivariate binomial (0,1) distributionapplication to 2×2 tables, Austral. J. Statist., 13, 154-158 (1971) · Zbl 0234.62011
[9] Hamdan, M. A.; Nasro, M. O., Maximum likelihood estimation of the parameters of the bivariate binomial distribution, Comm. Statist. A—Theory Methods, 15, 747-754 (1986) · Zbl 0601.62072
[10] Hamdan, M. A.; Tsokos, C. P., A model for physical and biological problemsthe bivariate-compounded Poisson distribution, Internat. Statist. Rev., 39, 59-63 (1971)
[11] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Discrete Multivariate Distributions (1997), Wiley: Wiley New York · Zbl 0868.62048
[12] Kocherlakota, S., A note on the bivariate binomial distribution, Statist. Probab. Lett., 8, 21-24 (1989) · Zbl 0667.62037
[13] Kocherlakota, S.; Kocherlakota, K., Bivariate Discrete Distributions (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0794.62002
[14] Le, C. T., A symmetric bivariate binomial distribution and its application to the analysis of clustered samples in medical research, Biometrical J., 26, 289-294 (1984)
[15] Ling, K. D.; Tai, T. H., On bivariate binomial distributions of order k, Soochow J. Math., 16, 211-220 (1990) · Zbl 0736.62043
[16] Loukas, S.; Kemp, C. D., The computer generation of bivariate binomial and negative binomial random variables, Comm. Statist. B—Simulation Comput., 15, 15-25 (1986) · Zbl 0617.65003
[17] Mardia, K. V., Families of Bivariate Distributions (1970), Griffin: Griffin London · Zbl 0223.62062
[18] Marshall, A. W.; Olkin, I., A family of bivariate distributions generated by the bivariate Bernoulli distribution, J. Amer. Statist. Assoc., 80, 332-338 (1985) · Zbl 0575.60023
[19] Mishra, A., A generalised bivariate binomial distribution applicable in four-fold sampling, Comm. Statist. A—Theory Methods, 25, 1943-1956 (1996) · Zbl 0925.62185
[20] Oluyede, B. O., A family of bivariate binomial distributions generated by extreme Bernoulli distributions, Comm. Statist. A—Theory Methods, 23, 1531-1547 (1994) · Zbl 0825.62146
[21] Ong, S. H., The computer generation of bivariate binomial variables with given marginals and correlation, Comm. Statist. B—Simulation Comput., 21, 285-299 (1992) · Zbl 0825.65004
[22] Papageorgiou, H.; David, K. M., On countable mixtures of bivariate binomial distributions, Biometrical J., 36, 581-601 (1994) · Zbl 0834.62044
[23] Patil, G. P.; Joshi, S. W., A Dictionary and Bibliography of Discrete Distribution (1968), Oliver and Boyd: Oliver and Boyd Edinburgh
[24] Pearson, K., Mathematical contribution to the theory of evolution VII. On the correlation of characters not quantitatively measurable, Philos. Trans., A195, 1-47 (1900) · JFM 32.0238.01
[25] Polson, N.; Wasserman, L., Prior distributions for the bivariate binomial, Biometrika, 77, 901-904 (1990)
[26] Szego, G., 1959. Orthogonal Polynomials. Colloquium Publication No. 23, American Mathematical Society, Providence, RI.; Szego, G., 1959. Orthogonal Polynomials. Colloquium Publication No. 23, American Mathematical Society, Providence, RI. · Zbl 0089.27501
[27] Takeuchi, K.; Takemura, A., On sum of 0-1 random variables (multivariate), Ann. Inst. Statist. Math., 39, 307-324 (1987) · Zbl 0658.60031
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.