Some general results on quantile functions for the generalized beta family
Abstract
In this article we study and obtain some general results on the quantile functions for the generalized beta family and the family of beta generated distributions. Having described the standardization rule, we have derived the quantile function of the 5 parameter generalized beta family of distributions [21, 22]. Further, quantile rules for distributional model building have been applied to generate quantile functions of several known and unknown distributions. Attempts have been made to obtain and study the quantile functions of size biased generalized beta distributions and generalized beta generated distributions. Finally, we have applied the proposed results to simulated as well as real datasets.References
A. Akinsete, and C. Lowe, Beta-rayleigh distribution in reliability measure. Section on physical and engineering sciences, Proceedings of the American Statistical Association, vol. 1, pp. 3103–3107, 2009.
A. Akinsete, F. Famoye, and C. Lee, The beta-pareto distribution Statistics, vol. 42, no. 6, pp. 547–563, 2008.
C. Alexander, G.M. Cordeiro, E.M.M. Ortega, and J.M. Sarabia, Generalized beta-generated distributions, Computational Statistics & Data Analysis, vol. 56, no. 6, pp. 1880–1897, 2012.
N. Balakrishnan, and H.Y. So, A generalization of quantile-based skew logistic distribution of van staden and king, Statistics & Probability Letters, vol. 107, pp. 44–51, 2015.
W. Barreto-Souza, G.M. Cordeiro, and A.B. Simas, Some results for beta fr´echet distribution, Communications in Statistics: Theory and Methods, vol. 40, no. 5, pp. 798–811, 2011.
G.M. Cordeiro, and M. De Castro, A new family of generalized distributions, Journal of statistical computation and simulation, vol. 81, no. 7, pp. 883–898, 2011.
Felipe RS De Gusm˜ao, E.M.M. Ortega, and G.M. Cordeiro, The generalized inverse weibul l distribution, Statistical Papers, vol. 52, no. 3, pp. 591–619, 2011.
F. Domma, and F. Condino, The beta-dagum distribution: definition and properties, Communications in Statistics: Theory and Methods, vol. 42, no. 22, pp. 4070–4090, 2013.
M.J. Ducey, and J.H. Gove, Size-biased distributions in the generalized beta distribution family, with applications to forestry, Forestry, vol. 81, no. 1, pp. 143–151, 2015.
N. Eugene, C. Lee, and F. Famoye, Beta-normal distribution and its applications, Communications in Statistics: Theory and methods, vol. 31, no. 4, pp. 497–512, 2002.
R.A. Fisher, The effect of methods of ascertainment upon the estimation of frequencies, Annals of Human Genetics, vol. 6, no. 1, pp. 13–25, 1934.
W.G. Gilchrist, Statistical model ling with quantile functions, Chapman and Hall, New York, 2000.
R.D. Gupta, and D. Kundu, Theory & methods: Generalized exponential distributions, Australian & New Zealand Journal of Statistics, vol. 41, no. 2, pp. 173–188, 1999.
R.D. Gupta, and D. Kundu, Exponentiated exponential family: an alternative to gamma and weibul l distributions, Biometrical journal, vol. 43, no. 1, pp. 117–130, 2001.
M.C. Jones, Families of distributions arising from distributions of order statistics, Test, vol. 13, no. 1, pp. 1–43, 2004.
J.K. Jose, and M Manoharan, Beta half logistic distribution-a new probability model for lifetime data, Journal of Statistics and Management Systems, vol. 19, no. 4, pp. 587–604, 2016.
M.S. Khan, G.R. Pasha, and A.H. Pasha, Theoretical analysis of inverse weibul l distribution, WSEAS Transactions on Mathematics, vol. 7, no. 2, pp. 30–38, 2008.
P. Kumaraswamy, A Generalized probability density function for double-bounded random processes, Journal of Hydrology, vol. 46, no. 1, pp. 79–88, 1980.
C. Lee, F. Famoye, and O. Olumolade, Beta-weibul l distribution: some properties and applications to censored data, Journal of modern applied statistical methods, vol. 6, no. 1, pp. 17, 2007.
A.J. Lemonte, W. Barreto-Souza, and G.M. Cordeiro, The exponentiated kumaraswamy distribution and its log- transform, Brazilian Journal of Probability and Statistics, vol. 27, no. 1, pp. 31–53, 2013.
J. B. McDonald, Some generalized functions for the size distribution of income, Econometrica, vol. 52, no. 3, pp. 647–663, 1984.
J.B. McDonald, and Y.J. Xu, A generalization of the beta distribution with applications, Journal of Econometrics, vol. 66, no. 1, pp. 133–152, 1995.
G.S. Mudholkar, D.K. Srivastava, and M. Freimer, The exponentiated weibul l family: a reanalysis of the bus-motor- failure data, Technometrics, vol. 37, no. 4, pp. 436–445, 1995.
S. Nadarajah, and A.K. Gupta, The beta fr´echet distribution, Far East Journal of Theoretical Statistics, vol. 14, no. 1, pp. 15–24, 2004.
S. Nadarajah, The exponentiated gumbel distribution with climate application, Environmetrics, vol. 17, no. 1, pp. 13–23, 2006.
S. Nadarajah, and S. Kotz, The beta gumbel distribution, Mathematical Problems in Engineering, vol. 2004, no. 4, pp. 323–332, 2004.
S. Nadarajah, and S. Kotz, The beta exponential distribution, Reliability engineering & system safety, vol. 91, no. 6, pp. 689–697, 2006.
S. Nadarajah, and S. Kotz, The exponentiated type distributions, Acta Applicandae Mathematica, vol. 92, no. 2, pp. 97–111, 2006.
P.F. Parana´ıba, E.M.M. Ortega, G.M. Cordeiro, and R.R. Pescim, The beta burr xii distribution with application to lifetime data, Computational Statistics & Data Analysis, vol. 55, no. 2, pp. 1118–1136, 2011.
E. Parzen, Quantile probability and statistical data modeling, Statistical Science, pp. 652–662, 2004.
C.R. Rao, On discrete distributions arising out of methods of ascertainment, In G.P. Patil, editor, Classical and Contagious Discrete Distributions, pp. 320–332. Pergamon Press and Statistical Publishing Society, Calcutta, 1965.
D. Sharma, and T.K. Chakrabarty, On size biased kumaraswamy distribution Statistics, Optimization and Information Computing, vol. 4, no. 3, pp. 252–264, 2016.
P.J. van Staden, and Robert A.R. King, The quantile-based skew logistic distribution, Statistics & Probability Letters, vol. 96, pp. 109–116, 2015.
W. Weibull, A statistical distribution function of wide applicability, Journal of Applied Mechanics-Transactions of the ASME, vol. 18, no. 3, pp. 293–297, 1951.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
