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A bivariate beta distribution.(English)Zbl 1116.60309

Summary: The Dirichlet distribution is often used as a prior distribution for the parameters of a multinomial distribution. Because this distribution has support on the simplex \(0 \leqslant x_i \leqslant 1\), \(\sum x_i=1\), it does not serve as the prior for a correlated binomial distribution. We here present a bivariate beta distribution that has support on \(0 \leqslant x_i \leqslant 1\), \(i=1,2\). When expanded in a power series it is related to the hypergeometric function. This bivariate density is positively likelihood ratio dependent and hence is positive quadrant dependent.

MSC:

60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions

Cite

References:

[1]Abramowitz, M., Stegun, I.A. (Eds.), 1964. Handbook of Mathematical Functions, Vol. 55. Applied Mathematics Series, US Department of Commerce, National Bureau of Standards, Washington, DC.; Abramowitz, M., Stegun, I.A. (Eds.), 1964. Handbook of Mathematical Functions, Vol. 55. Applied Mathematics Series, US Department of Commerce, National Bureau of Standards, Washington, DC. ·Zbl 0171.38503
[2]Erdélyi, A. (Ed.), 1953. Higher Transcendental Functions, Vol. 1. McGraw-Hill, New York.; Erdélyi, A. (Ed.), 1953. Higher Transcendental Functions, Vol. 1. McGraw-Hill, New York. ·Zbl 0051.30303
[3]Kotz, S., Balakrishnan, N., Johnson, N.L., 2000. Continuous Multivariate Distributions, 2nd Edition, Vol. 1. Wiley, New York.; Kotz, S., Balakrishnan, N., Johnson, N.L., 2000. Continuous Multivariate Distributions, 2nd Edition, Vol. 1. Wiley, New York. ·Zbl 0946.62001
[4]Sobel, M., Uppuluri, V.R.R., Frankowski, K., 1977. Dirichlet Distribution—Type1. In: Owen, D.B., Odeh, R.E., Davenport, J.M. (Eds.), Selected Tables in Mathematical Statistics, Vol. 4. American Mathematical Society, Providence, RI.; Sobel, M., Uppuluri, V.R.R., Frankowski, K., 1977. Dirichlet Distribution—Type1. In: Owen, D.B., Odeh, R.E., Davenport, J.M. (Eds.), Selected Tables in Mathematical Statistics, Vol. 4. American Mathematical Society, Providence, RI. ·Zbl 0356.62020
[5]Sobel, M., Uppuluri, V.R.R., Frankowski, K., 1985. Dirichlet integrals of type 2 and their applications. In: Odeh, R.E., Davenport, J.M., Pearson, N.S. (Eds.), Selected Tables in Mathematical Statistics, Vol. 9. American Mathematical Society, Providence, RI.; Sobel, M., Uppuluri, V.R.R., Frankowski, K., 1985. Dirichlet integrals of type 2 and their applications. In: Odeh, R.E., Davenport, J.M., Pearson, N.S. (Eds.), Selected Tables in Mathematical Statistics, Vol. 9. American Mathematical Society, Providence, RI. ·Zbl 0586.62197
[6]Tong, Y. L., Probability Inequalities in Multivariate Distributions (1980), Academic Press: Academic Press New York ·Zbl 0455.60003
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.
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