The Weibull Birnbaum-Saunders Distribution And Its Applications
Abstract
A new lifetime model, with four positive parameters, called the Weibull Birnbaum-Saunders distribution is proposed. The proposed model extends the Birnbaum-Saunders distribution and provides great flexibility in modeling data in practice. Some mathematical properties of the new distribution are obtained including expansions for the cumulative and density functions, moments, generating function, mean deviations, order statistics and reliability. Estimation of the model parameters is carried out by the maximum likelihood estimation method. A simulation study is presented to show the performance of the maximum likelihood estimates of the model parameters. The flexibility of the new model is examined by applying it to two real data sets.References
A. Z. Afify, G. M. Cordeiro, E. M. M. Ortega, H. M. Yousof and N. M. Butt, The four-parameter Burr XII distribution: Properties, regression model, and applications, Communications in Statistics-Theory and Methods, vol. 47, pp. 2605-2624, 2018.
S. A. Bakar, N. Hamzah, M. Maghsoudi and S. Nadarajah, Modeling loss data using composite models, Insurance: Mathematics and Economics, vol. 12, pp. 146–154, 2015.
N. Balakrishnan and D. Kundu, Birnbaum-Saunders distribution: A review of models, analysis, and applications, Applied Stochastic Models in Business and Industry, vol. 35, no. 1, pp. 4–49, 2019.
M. Barros, M. Galea, V. Leiva and M. Santos-Neto, Generalized Tobit models: Diagnostics and application in econometrics, Journal of Applied Statistics, vol. 45, pp. 145–167, 2018.
Z. W. Birnbaum and S. C. Saunders, A new family of life distributions, Journal of Applied Probability, vol. 6, pp. 319–327, 1969.
M. Bourguignon, J. Leao, V. Leiva and M. Santos-Neto, The transmuted Birnbaum-Saunders distribution, Revstat Statistical Journal, vol. 15, pp. 601–628, 2017.
M. Bourguignon, R. B. Silva and G. M. Cordeiro, The Weibull–G family of probability distributions, Journal of Data Science, vol. 12, pp. 53–68, 2014.
G. M. Cordeiro and A.J. Lemonte, The β-Birnbaum-Saunders distribution: An improved distribution for fatigue life modeling, Computational Statistics and Data Analysis, vol. 55, pp. 1445–1461, 2011.
G. M. Cordeiro and A.J. Lemonte, The exponentiated generalized Birnbaum-Saunders distribution, Applied Mathematics and Computation, vol. 247, pp. 762–779, 2014.
G. M. Cordeiro, A.J. Lemonte and E. M. M. Ortega, An extended fatigue life distribution, Statistics, vol. 47, pp. 626–653, 2013.
G. M. Cordeiro, M. C. D. S. Lima, A. H. M. A. Cysneiros, M. A. R. Pascoa, R. R. Pescim and E. M. M. Ortega, An extended
Birnbaum–Saunders distribution: Theory, estimation, and applications, Communications in Statistics - Theory and Methods, vol.45, pp. 2268–2297, 2016.
D. R. Cox and D. V. Hinkley, Theoretical statistics, Chapman and Hall, London, 1974.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, New York: Academic Press, 2007.
R. D. Gupta and D. Kundu, Exponentiated exponential family: an alternative to gamma and Weibull distributions, Biometrical Journal, vol. 43, pp. 117–130, 2001.
F. Hashemi, M. Naderi and M. Mashinchi, Clustering right-skewed data stream via Birnbaum–Saunders mixture models: A flexible approach based on fuzzy clustering algorithm, Applied Soft Computing, vol. 82, Article 105539, 2019.
F. Hashemi, M. Naderi and A. Jamalizadeh, Normal mean-variance Lindley Birnbaum–Saunders distribution, Statistics and Its Interface, vol. 12, no. 4, pp. 585–597, 2019.
A. Jamalizadeh, F. Hashemi and M. Naderi, Discussion of ”Birnbaum-Saunders distribution: A review of models, analysis, and applications”, Applied Stochastic Models in Business and Industry, vol. 35, no. 1, pp. 82–89, 2019.
V. Leiva, The Birnbaum–Saunders distribution, Academic Press, 2016.
V. Leiva, M. Barros, G. A. Paula and M. Galea, Influence diagnostics in log-Birnbaum–Saunders regression models with censored data, Computational Statistics and Data Analysis, vol. 51, pp. 5694–5707, 2007.
A. J. Lemonte, A new extension of the Birnbaum-Saunders distribution, Brazilian Journal of Probability and Statistics, vol. 27, pp. 133–149, 2013.
K. Mohammadi, O. Alavi and J. G. McGowan, Use of Birnbaum-Saunders distribution for estimating wind speed and wind power probability distributions: A review, Energy Conversion and Management, vol. 143, pp. 109–122, 2017.
W. Q. Meeker and L. A. Escobar, Statistical methods for reliability data, John Wiley and Sons, New York, 1998.
G. S. Mudholkar and D. K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure real data, IEEE Transactions on Reliability, vol. 42, pp. 299–302, 1993.
M. Naderi, F. Hashemi, A. Bekker and A. Jamalizadeh, Modeling right-skewed financial data streams: A likelihood inference based on the generalized Birnbaum–Saunders mixture model, Applied Mathematics and Computation, vol. 376, Article 125109, 2020.
M. Naderi, M.Mozafari and K. Okhli, Finite mixture modeling via skew-Laplace Birnbaum–Saunders distribution, Journal of Statistical Theory and Applications, vol. 19, pp. 49–58, 2020.
G. A. Paula, V. Leiva, M. Barros and S. Liu, Robust statistical modeling using the Birnbaum-Saunders-t distribution applied to insurance, Applied Stochastic Models in Business and Industry, vol. 28, no. 1, pp. 16–34, 2011.
J. R. Rieck, A moment-generating function with application to the Birnbaum-Saunders distribution, Communications in Statistics-Theory and Methods, vol. 28, pp. 2213–2222, 1999.
A. M. Sarhan and J. Apaloo, Exponentiated modified Weibull extension distribution, Reliability Engineering and System Safety, vol. 112, pp. 137–144, 2013.
H. Saulo, J. Le˜ ao and M. Bourguignon, The Kumaraswamy Birnbaum–Saunders distribution, Journal of Statistical Theory and Practice, vol. 6, pp. 745–759, 2012.
M. H. Tahir, G. M. Cordeiro, A. Alzaatreh, M. Mansoor and M. Zubair, A new Weibull-Pareto distribution: properties and applications, Communications in Statistics - Simulation and Computation, vol. 45, no. 10, pp. 3548–3567, 2016.
M. H. Tahir, G. M. Cordeiro, M. Mansoor and M. Zubair, The Weibull-Lomax distribution: properties and applications, Hacettepe Journal of Mathematics and Statistics, vol. 44, pp. 455–474. 2015.
M.H.Tahir,G.M.Cordeiro,M.Mansoor,M.Zubair and M.Alizadeh, The Weibull-Dagum distribution: properties and applications, Communications in Statistics-Theory and Methods, vol. 45, pp. 7376–7398, 2016.
R. Terras, A Miller algorithm for an incomplete Bessel function, Journal of Computational Physics, vol. 39, pp. 233–240, 1981.
K. Xu, M. Xie, L. C. Tang and S. L. Ho, Application of neural networks in forecasting engine systems reliability, Applied Soft Computing, vol. 2, pp. 255–268, 2003.
- 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).
