The beta Nadarajah-Haghighi distribution

Abstract

Recently, there has been a great interest among statisticians and applied researchers in constructing flexible distributions for better modeling

non-monotone failure rates. We study a lifetime model of the beta generated family, called the beta Nadarajah-Haghighi distribution, which can be used to model survival data. The proposed model includes as special models some important distributions. The hazard rate function is an important quantity characterizing life phenomena. Its hazard function can be constant, decreasing, increasing, upsidedown bathtub and bathtub-shaped depending on the parameters. We provide a comprehensive mathematical treatment of the new distribution and derive explicit expressions for some of its basic mathematical quantities. The method of maximum likelihood is used for estimating the model parameters and a small Monte Carlo simulation is conducted. We fit the proposed model to two real data sets to prove empirically its flexibility as compared to other lifetime distributions 

Keywords

• Beta distribution,Moment,Nadarajah-Haghighi distribution,Mean deviation

References

1. Akinsete, A., Famoye, F. & Lee, C. The beta-Pareto distribution. Statistics 42(6), 547-563, 2008.
2. Barreto-Souza, W., Santos, A.H.S. & Cordeiro, G.M. The beta generalized exponential distribution. Journal of Statistical Computation and Simulation, 80(2), 159-172, 2010.
3. Chen, G., Balakrishnan, N. A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, 154-161, 1995.
4. Cordeiro, G. M., Lemonte, A. J. The Beta-Birnbaum-Saunders distribution: An improved distribution for fatigue life modeling. Computational Statistics and Data Analysis, 55, 1445- 1461, 2011.
5. Doornik, J. A. Ox 6: Object-oriented matrix programming language. 5th ed. London: Timberlake Consultants, 2009.
6. Eugene, N., Lee, C. & Famoye, F. Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31, 497-512, 2002.
7. Famoye F, Lee C, Olumolade O. The beta-Weibull distribution. Journal of Statistical Theory and Applications, 54, 121-36, 2008.
8. Gradshteyn, I. S., Ryzhik, I. M. Table of Integrals, Series, and Products. Edited by Alan Jeffrey and Daniel Zwillinger, 7th edn, Academic Press, New York, 2007.

9. Gupta, R.C., Gupta, P.L. & Gupta R.D. Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and Methods,27, 887-904, 1998.
10. Gupta, R.D. & Kundu, D. Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41, 173-188, 1999.
11. Jones,M.C. Families of distributions arising from distributions of order statistics (with discussion). Test, 13, 1-43, 2004.
12. Kenney, J. F., Keeping, E. S. Mathematics of Statistics (3rd ed.), Part 1. New Jersey, 1962.
13. Lemonte, A. A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics and Data Analysis, 62, 149-170, 2013.
14. Lee, E.T. Statistical Methods for Survival Data Analysis. Wiley: New York, 1992.
15. Lee, E.T., Wang, J.W.Statistical Methods for Survival Data Analysis, third ed. Wiley, New York, 2003.
16. Lee, C., Famoye, F. & Olumolade, O. Beta-Weibull distribution: Some properties and applications to censored data. Journal of Modern Applied Statistical Methods, 6(1), 173-186, 2007.
17. Moors, J. J. A. A quantile alternative for kurtosis. Journal of the Royal Statistical Society (Series D), 37, 25-32, 1988.
18. Mudholkar, G.S., Srivastava, D.K. & Freimer, M. The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics, 37, 436-445, 1995.
19. Nadarajah, S. The exponentiated Gumbel distribution with climate application. Environmetrics, 17, 13-23, 2005.
20. Nadarajah, S., Haghighi, F. An extension of the exponential distribution. Statistics: A Journal of Theoretical and Applied Statistics, 45, 543-558, 2011.
21. Nadarajah, S., Kotz, S. The beta Gumbel distribution. Mathematical Problems in Engineering, 10, 323-332, 2004.
22. Nadarajah, S., Gupta, A.K. The beta Fréchet distribution. Far East Journal of Theoretical Statistics, 14, 15-24, 2004.
23. Nadarajah, S. & Kotz, S. The beta exponential distribution. Reliability Engineering System Safety, 91, 689-697, 2006.
24. Pescim, R.R., Demétrio, C.G.B., Cordeiro, G.M., Ortega, E.M.M. & Urbano, M.R., 2010. The beta generalized half-normal distribution. Computational Statistics and Data Analysis, 54, 945-957, 2010.
25. Rényi, A., 1961. On measures of information and entropy. Proceedings of the fourth Berkeley Symposium on Mathematics, Statistics and Probability, 1960, 547-561, 1961.
26. Rigby, R. A., Stasinopoulos D. M. Generalized additive models for location, scale and sahpe. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54, 507-554, 2005.
27. Shannon, C.E. Prediction and entropy of printed English. The Bell System Technical Journal, 30, 50-64, 1951.