Generalized Dirichlet Distribution Based on Confluent Hypergeometric Series
DOI:
https://doi.org/10.54097/ajst.v5i2.6461Keywords:
Dirichlet distribution, Confluent hypergeometric function, Gamma distribution, Non-central gamma distribution.Abstract
Dirichlet distribution is a kind of high-dimensional continuous probability distribution, which has important applications in the fields of statistics, machine learning and bioinformatics. In this paper, based on gamma distribution we study two two-dimensional random variables. Then we derive the properties of these two two-dimensional random variables by using the properties of non-central gamma distribution and confluent hypergeometric series. From these properties, we find the two random variables follow generalized Dirichlet distributions. Applying hypergeometric series to Dirichlet distribution broadens the research of Dirichlet distribution.
Downloads
References
Kai Wang Ng, Guo-Liang Tian, Man-Lai Tang. Dirichlet and Related Distributions: Theory, Methods and Applications. New York: John Wiley & Sons, 2011.
Mohamed Maher Ben Ismaila, Hichem Friguib. Unsupervised clustering and feature weighting based on Generalized Dirichlet mixture modeling. Information Sciences, 2014, 274: 35-54.
David M. Blei, Andrew Y. Ng, Michael I. Jordan. Latent Dirichlet Allocation. Journal of Machine Learning Research, 2003, 3: 993-1022.
S. Shivashankar, S. Srivathsan, B. Ravindran, Ashish V. Tendulkar. Multi-view methods for protein structure comparison using latent dirichlet allocation. Bioinformatics, 2011, 27: i61-i68.
Arjun K. Gupta, Johanna Marcela Orozco-Castañeda, Daya K. Nagar. Non-central bivariate beta distribution. Springer, 2011, 52: 139-152.
Johanna Marcela Orozco-Castañeda, Daya K. Nagar, Arjun K. Gupta. Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind. Computers and Mathematics with Applications, 2012, 64: 2507-2519.
Yudell L. Luke. The Special Functions and Their Approximations. Mathematics in Science and Engineering, 1969,53: 1-349.
D. K. Nagar, Johanna Marcela Orozco-Castañeda, Arjun K. Gupta. Product and quotient of correlated beta variables. Appl. Math. Lett., 2009, 22: 105-109.
Ingram Olkin, Ruixue Liu. A bivariate beta distribution. Statistics & Probability Letters, 2003, 62: 407-412.
Saralees Nadarajah, Samuel Kotz. Some bivariate beta distributions. Statistics, 2005, 39: 457-466.
James J. Chen, Melvin R. Novick, Bayesian Analysis for Binomial Models with Generalized Beta Prior Distributions. Journal of Educational and Behavioral Statistics, 1984, 9: 163-175.
James M. Dickey. Multiple Hypergeometric Functions: Probabilistic Interpretations and Statistical Uses.Journal of the American Statistical Association, 1983, 78: 628-637.
Mohamed Abdalla, Muajebah Hidan. Analytical properties of the two-variables Jacobi matrix polynomials with applications. Demonstratio Mathematica, 2021, 54: 178-188.
Daya K. Nagar, Fabio Fabio Sepúlveda-Murillo. Properties of confluent hypergeometric function kind 1 distribution. Journal of Interdisciplinary Mathematics, 2008, 11: 495-504.
Jiayu Lin. On The Dirichlet Distribution by Jiayu Lin. Canada: Mathematics, 2016.
A. M. Mathai, R. K. Saxena. On a generalized hypergeometric distribution. Springer, 1967, 11: 127-132.
W. N. Bailey. Generalized hypergeometric series .London: Mathematics, 1953.
Wentao Fan, Hassen Sallay, Nizar Bouguila. Online Learning of Hierarchical Pitman-Yor Process Mixture of Generalized Dirichlet Distributions With Feature Selection. IEEE Trans Neural Netw Learn Syst, 2017, 28: 2048-2061.
