On the k – Higher-Order Beta Distribution Function

Benson Ade Afere

doi:https://doi.org/10.14445/22315373/IJMTT-V66I11P505

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 11 | Year 2020 | Article Id. IJMTT-V66I11P505 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I11P505

On the k – Higher-Order Beta Distribution Function

Benson Ade Afere

Citation :

Benson Ade Afere, "On the k – Higher-Order Beta Distribution Function," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 11, pp. 67-79, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I11P505

Abstract

In this work, a new higher-order beta probability distribution function is proposed from the existing beta probability distribution function. The new probability distribution function was derived using the work of [5]. The properties of the two distribution functions were given. A Monte Carlo experiment is performed for two scenarios using small and large sample sizes and it was observed that the proposed distribution has the least mean square error. It was equally observed that for the two scenarios, as the sample size increases, the error decreases which obey the finite sample theory. More importantly, based on the observations, the proposed distribution is efficient even if the data set departs from the standard beta distribution. A real life applications were used to stress further the flexibility of the proposed distribution.

Keywords

Probability distribution, classical beta distribution function, higher-order beta distribution function, mean square error, Monte Carlo experiment.

References

[1] K. M. Aludant(2018), On the Beta Cumulative Distribution Function, Applied Mathematical Sciences, Vol. 12, No. 10, 461 – 466.
[2] A. Alzaatreh, C. Lee and F. Famoye(2014),T-normal family of distributions: a new approach to generate the normal distribution, Journal of Statistical Distribution and Applications,1:6,1-8, http://www.jsdajournal.com/content/1/1/16
[3] D. F. Andrews and A. M. Herzberg(1985), Data: A Collection of Problems from Many Fields for the Student and Research Worker (Springer Series in Statistics), Springer, New York.
[4] N. Eugene, C.Lee and F. Famoye(2002), Beta-Normal Distribution and its Applications, Communication Statistical Theory Method, Vol. 31(4), 497 – 512.
[5] M. C. Jones and P. J. Foster (1993), Generalized Jackknifing and higher-order kernels, Journal of Nonparametric Statistics 3, 81 – 94.
[6] S. Kotz and D. Vicari (2005), Survey of developments in the theory of continuous skewed distributions, Metron, 63, 225 – 261.
[7] C. Lee, F. Famoye and A. Y. Alzaatreh (2013), Methods for generating families of univariate continuous distributions in the recent decades, WIREs Computational Statistics, 5, 219 – 238.
[8] J. E. Lighter (1991), A Brief Look at the History of Probability and Statistics, The Mathematics Teacher, Vol. 84, No. 8, 623 – 630.
[9] C. E. B. Owen (2008), Parameter estimation for the beta distribution, All Theses and Dissertations-1614, https://scholararchive-byu.edu/etd/1614.
[10] G. Rahman, S. Mubeen, A. Rehman and M. Naz (2014), On k-Gamma and k-Beta Distribution and Moments Generating Functions, Journal of Probability and Statistics, Vol. 2014, Article ID:982013, 6 pages, http://dx.doi.org/10.1155/2014/982013.
[11] D. D. Wackerly, W. Mendenhall III and R. L.Scheaffer (2008), Mathematical Statistics with Applications, The Thomson Corporation, USA.