Theory and Applications of the Transmuted Continuous Bernoulli Distribution
Abstract
Modern applied statistics naturally give rise to the continuous Bernoulli distribution (data fitting, deep learning, computer vision, etc). On the mathematical side, it can be viewed as a one-parameter distribution corresponding to a special exponential distribution restricted to the unit interval. As a matter of fact, manageable extensions of this distribution have great potential in the same fields. In this study, we motivate a transmuted version of the continuous Bernoulli distribution with the goal of analyzing proportional data sets. The feature of the created transmuted continuous Bernoulli distribution is an additional parameter that realizes a linear tradeoff between the min and max of two continuous random variables with the continuous Bernoulli distribution. The standard study process is respected: we derive some mathematical properties of the proposed distribution and adopt the maximum likelihood estimation technique in estimating the unknown parameters involved. A Monte Carlo simulation exercise was conducted to examine and confirm the asymptotic behavior of the obtained estimates. In order to show the applicability of the proposed distribution, three proportional data sets are analyzed and the results obtained are compared with competitive distributions. Empirical findings reveal that the transmuted continuous Bernoulli distribution promises more flexibility in fitting proportional data sets than its competitors.
References
G. R. Aryal and C. P. Tsokos, Transmuted Weibull distribution : A generalization of the Weibull probability distribution, European Journal of Pure and Applied Mathematics 4(2) (2011), 89-102.
R. A. R. Bantan, C. Chesneau, F. Jamal, M. Elgarhy, M. H. Tahir, A. Aqib, M. Zubair and S. Anam, Some new facts about the unit-Rayleigh distribution with applications, Mathematics 8(11) (2020), 1954. https://doi.org/10.3390/math8111954
M. Bourguignon, I. Ghosh and G. M. Cordeiro, General results for the transmuted family of distributions and new models, Journal of Probability and Statistics 2016 (2016), Article ID 7208425, 12 pp. https://doi.org/10.1155/2016/7208425
G. Casella and R. L. Berger, Statistical Inference, Brooks/Cole Publishing Company : Bel Air, CA, USA, 1990.
V. Choulakian, and M. A. Stephens, Goodness-of-fit for the generalized Pareto distribution, Technometrics 43(4) (2001), 478-484.
I. Elbatal, Transmuted modified inverse Weibull distribution, International Journal of Mathematical Archive 4(8) (2013), 117-129. https://doi.org/10.12988/ams.2014.44267
M. Elgarhy, M. Rashed and A. W. Shawki, Transmuted generalized Lindley distribution, International Journal of Mathematics Trends and Technology 29(2) (2016), 145-154. https://doi.org/10.14445/22315373/IJMTT-V29P520
M. Ghitany, B. Atieh and S. Nadarajah, Lindley distribution and its applications, Mathematics and Computers in Simulation 78 (2008), 493-506. https://doi.org/10.1016/j.matcom. 2007.06.007
E. Gordon-Rodriguez, G. Loaiza-Ganem and J. P. Cunningham, The continuous categorical : a novel simplex-valued exponential family, 36th International Conference on Machine Learning, ICML 2020, International Machine Learning Society (IMLS), 2020.
S. A. Kemaloglu and M. Yilmaz, Transmuted two-parameter Lindley distribution, Communications in Statistics Theory and Methods 46(23) (2017), 11866-11879. https://doi.org/10.1080/03610926.2017.1285933
J. P. Klein and M. L. Moeschberger, Survival Analysis : Techniques for Censored and Truncated Data, Springer, Berlin, 2006.
M. Korkmaz and C. Chesneau, On the unit Burr-XII distribution with the quantile regression modeling and applications, Computational and Applied Mathematics 40(1) (2021), 1-26. https://doi.org/10.1007/s40314-021-01418-5
G. Loaiza-Ganem and J. P. Cunningham, The continuous Bernoulli : fixing a pervasive error in variational autoencoders, Advances in Neural Information Processing Systems (2019), 13266-13276.
F. Merovci, Transmuted Lindley distribution, International Journal of Open Problems in Computer Science and Mathematics 6(2) (2013), 63-72. https://doi.org/10.12816/0006170
K. Modi and V. Gill, Unit Burr-III distribution with application, Journal of Statistics and Management Systems 23 (2020), 579-592. https://doi.org/10.1080/09720510.2019.1646503
F. C. Opone, and B. N. Iwerumor, A new Marshall-Olkin extended family of distributions with bounded support, Gazi University Journal of Science 34(3) (2021), 899-914. https://doi.org/10.35378/gujs.721816
F. C. Opone and J. E. Osemwenkhae, The transmuted Marshall-Olkin extended Topp-Leone distribution, Earthline Journal of Mathematical Sciences 9(2) (2022), 179-199. https://doi.org/10.34198/ejms.9222.179199
W. T. Shaw and I. R. C. Buckley, The alchemy of probability distributions : beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map, UCL Discovery Repository, 2007. http://discovery.ucl.ac.uk/id/eprint/643923
K. Wang and M. Lee, Continuous Bernoulli distribution : simulator and test statistic, 2020. http://dx.doi.org/10.13140/RG.2.2.28869.27365
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
