Bivariate Selberg-beta type 1 distribution. (English) Zbl 1503.62048
Summary: By using the well known Selberg integral we define a bivariate beta distribution. Unlike other bivariate beta distributions which have support on the simplex \(\{(x, y) : x > 0,\, y > 0, \,x + y < 1\}\) this distribution has support on \(\{(x, y) : 0 < x < 1, \, 0 < y < 1\}\). Several properties such as marginal and conditional distributions, joint moments, coefficient of correlation, entropies, Fisher information matrix and estimation of parameters have been studied.
Keywords:
beta distribution; bivariate distribution; entropy; hypergeometric function; moments; Selberg integral; transformationReferences:
| [1] | Borsos, Ákos; Lakatos, Béla G., Simulation and analysis of crystallization of high aspect ratio crystals with fragmentation, Comput Aided Chem Eng, 30, 1028-1032 (2012) |
| [2] | Alshkaki, Rafid S. A., A six parameters beta distribution with application for modeling waiting time of muslim early morning prayer, Ann Data Sci, 8, 57-90 (2021) |
| [3] | Armero, C.; Bayarri, M., Prior assessments for predictions in queues, Statistician, 43, 1, 139-153 (1994) |
| [4] | Bury, Karl, Statistical distributions in engineering (1999), Cambridge University Press: Cambridge University Press Cambridge |
| [5] | Gómez-Déniz, Emilio; Sarabia, José María, A family of generalised beta distributions: properties and applications, Ann Data Sci, 5, 401-420 (2018) |
| [6] | Gordy, M., Computationally convenient distributional assumptions for common-value auctions, Comput Econ, 12, 61-78 (1998) · Zbl 0912.90093 |
| [7] | Gupta, A. K.; Nagar, D. K., Properties of matrix variate beta type 3 distribution, Int J Math Math Sci, 2009, Article 308518 pp. (2009), 18 · Zbl 1177.60016 |
| [8] | Libby, D. L.; Novic, M. R., Multivariate generalized beta distributions with applications to utility assessment, J Educ Stat, 7, 4, 271-294 (1982) |
| [9] | McDonald, James; Xu, Yexiao J., A generalization of the beta distribution with applications, J Econometrics, 69, 2, 427-428 (1995) · Zbl 0833.62012 |
| [10] | Nadarajah, Saralees; Kotz, Samuel, An \(F_1\) beta distribution with bathtub failure rate function, Amer J Math Management Sci, 26, 1-2, 113-131 (2006) · Zbl 1154.62308 |
| [11] | Pham-Gia, T.; Duong, Q. P., The generalized beta- and \(F\)-distributions in statistical modelling, Math Comput Model, 12, 12, 1613-1625 (1989) · Zbl 0697.62010 |
| [12] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous univariate distributions-2 (1994), John Wiley & Sons: John Wiley & Sons New York · Zbl 0811.62001 |
| [13] | Nadarajah, Saralees; Kotz, Samuel, Multitude of beta distributions with applications, Statistics, 41, 2, 153-179 (2007) · Zbl 1117.62018 |
| [14] | (Gupta, Arjun K.; Nadarajah, Saralees, Handbook of beta distribution and its applications. Handbook of beta distribution and its applications, Statistics: textbooks and monographs, vol. 174 (2004), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York) · Zbl 1062.62021 |
| [15] | Ghosh, Indranil, On the reliability for some bivariate dependent beta and kumaraswamy distributions: a brief survey, Stoch Q Control, 34, 2, 115-121 (2019) · Zbl 1432.62345 |
| [16] | Nadarajah, Saralees; Kotz, Samuel, Some bivariate beta distributions, Statistics, 39, 5, 457-466 (2005) · Zbl 1089.62012 |
| [17] | Gupta, Rameshwar D.; Richards, Donald St. P., The history of the Dirichlet and Liouville distributions, Int Stat Rev, 69, 3, 433-446 (2001) · Zbl 1171.60301 |
| [18] | Connor, Robert J.; Mosimann, James E., Concepts of independence for proportions with a generalization of the Dirichlet distribution, J Amer Statist Assoc, 64, 325, 194-206 (1969) · Zbl 0179.24101 |
| [19] | Chen, J. C.; Novic, M. R., Bayesian analysis for binomial models with generalized beta prior distribution, J Educ Stat, 7, 2, 163-175 (1984) |
| [20] | Gupta, A. K.; Orozco-Castañeda, J. M.; Nagar, D. K., Non-central bivariate beta distribution, Statist Papers, 52, 139-152 (2011) · Zbl 1247.62039 |
| [21] | Nagar, Daya K.; Rada-Mora, Erika Alejandra, Properties of multivariate beta distributions, Far East J Theor Stat, 24, 1, 73-94 (2008) · Zbl 1140.62323 |
| [22] | Olkin, I.; Liu, R., A bivariate beta distribution, Statist Probab Lett, 62, 407-412 (2003) · Zbl 1116.60309 |
| [23] | Orozco-Castañeda, Johanna Marcela; Nagar, Daya K.; Gupta, Arjun K., Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind, Comput Math Appl, 64, 8, 2507-2519 (2012) · Zbl 1268.60016 |
| [24] | Nadarajah, Saralees, The bivariate \(F_3\)-beta distribution, Commun Korean Math Soc, 21, 2, 363-374 (2006) · Zbl 1160.33308 |
| [25] | Nadarajah, Saralees, The bivariate \(F_2\)-beta distribution, Amer J Math Management Sci, 27, 3-4, 351-368 (2007) · Zbl 1154.62340 |
| [26] | Nadarajah, Saralees; Kotz, Samuel, The bivariate \(F_1\)-beta distribution, C R Math L’Acad Sci, 27, 2, 58-64 (2005) · Zbl 1079.33011 |
| [27] | Nadarajah, Saralees; Shih, Shou Hsing; Nagar, Daya K., A new bivariate beta distribution, Statistics, 51, 2, 455-474 (2017) · Zbl 1369.62028 |
| [28] | Sarabia, José María; Castillo, Enrique, Bivariate distributions based on the generalized three-parameter beta distribution, advances in distribution theory, order statistics, and inference, (Balakrishnan, N.; Sarabia, J. M.; Castillo, E., Statistics for industry and technology (2006), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA), 85-110 · Zbl 05196665 |
| [29] | Bran-Cardona, Paula Andrea Johanna Marcela Orozco-Castañeda; Nagar, Daya Krishna, Bivariate generalization of the kummer-beta distribution, Rev Colombiana Estadística, 34, 3, 497-512 (2011) · Zbl 1266.60011 |
| [30] | Nagar, Daya K.; Nadarajah, Saralees; Okorie, Idika E., A new bivariate distribution with one marginal defined on the unit interval, Ann Data Sci, 4, 3, 405-420 (2017) |
| [31] | Balakrishnan, N.; Lai, Chin-Diew, Continuous bivariate distributions (2009), Springer: Springer Dordrecht · Zbl 1267.62028 |
| [32] | Arnold, Barry C.; Castillo, Enrique; Sarabia, José María, (Conditional specification of statistical models. Conditional specification of statistical models, Springer series in statistics (1999), Springer-Verlag: Springer-Verlag New York) · Zbl 0932.62001 |
| [33] | Kotz, Samuel; Balakrishnan, N.; Johnson, Norman L., (Continuous multivariate distributions. 1. models and applications. Continuous multivariate distributions. 1. models and applications, Wiley series in probability and statistics: applied probability and statistics (2000), Wiley-Interscience: Wiley-Interscience New York) · Zbl 0946.62001 |
| [34] | Gupta, A. K.; Nagar, D. K., Matrix variate distributions (2000), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · Zbl 0935.62064 |
| [35] | Nagar, Daya K.; Roldán-Correa, Alejandro; Gupta, Arjun K., Bimatrix variate kummer-gamma distribution, Results Appl Math, 11, Article 100161 pp. (2021), 7 · Zbl 1472.62075 |
| [36] | Nagar, Daya K.; Naranjo-Ríos, Sandra Milena; Gupta, Arjun K., Evaluation of a generalized selberg integral, Random Oper Stoch Equ, 23, 1, 11-20 (2015) · Zbl 1309.33007 |
| [37] | Shannon, C. E., A mathematical theory of communication, Bell Syst Tech J, 27, 379-423 (1948), 623-656 · Zbl 1154.94303 |
| [38] | Rényi, A., On measures of entropy and information, (Proceedings of the fourth berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics (1961), University of California Press: University of California Press Berkeley, California), 547-561 · Zbl 0106.33001 |
| [39] | Nadarajah, S.; Zografos, K., Expressions for Rényi and Shannon entropies for bivariate distributions, Inform Sci, 170, 2-4, 173-189 (2005) · Zbl 1066.62007 |
| [40] | Zografos, K., On maximum entropy characterization of pearson’s type II and VII multivariate distributions, J Multivariate Anal, 71, 1, 67-75 (1999) · Zbl 0951.62040 |
| [41] | Zografos, K.; Nadarajah, S., Expressions for Rényi and Shannon entropies for multivariate distributions, Statist Probab Lett, 71, 1, 71-84 (2005) · Zbl 1058.62008 |
| [42] | Nagar, Daya K.; Orozco-Castañeda, Johanna Marcela; Gupta, Arjun K., Product and quotient of correlated beta variables, Appl Math Lett, 22, 1, 105-109 (2009) · Zbl 1163.60302 |
| [43] | Johnson, Norman L.; Kemp, Adrienne W.; Kotz, Samuel, (Univariate discrete distributions. Univariate discrete distributions, Wiley series in probability and statistics (2005), Wiley-Interscience [John Wiley & Sons]: Wiley-Interscience [John Wiley & Sons] Hoboken, NJ) · Zbl 1092.62010 |
| [44] | Bibby, Bo Martin; Michael Væth, T., The two-dimensional beta binomial distribution, Stat Probab Lett, 81, 7, 884-891 (2011) · Zbl 1219.62020 |
| [45] | Scollnik, David Peter Michael, Bayesian inference for three bivariate beta binomial models, Open Stat Probab J, 08, 27-38 (2017) |
| [46] | Luke, Y. L., The special functions and their approximations, Vol. 1 (1969), Academic Press: Academic Press New York · Zbl 0193.01701 |
| [47] | Srivastava, H. M.; Karlsson, P. W., Multiple gaussian hypergeometric series (1985), John Wiley & Sons: John Wiley & Sons New York · Zbl 0552.33001 |
| [48] | Bailey, W. N., Generalized hypergeometric series (1964), Stechert-Hafner Service Agency: Stechert-Hafner Service Agency New York · JFM 61.0406.01 |
| [49] | Andrews, George E.; Askey, Richard; Roy, Ranjan, (Special functions. Special functions, Encyclopedia of mathematics and its applications, vol. 71 (1999), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0920.33001 |
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