Modified slash Birnbaum-Saunders distribution

Abstract

In this paper, we introduce an extension for the Birnbaum-Saunders (BS) distribution based on the modified slash (MS) distribution proposed
by [12]. This new family of BS type distributions is obtained by replacing the usual normal distribution with the quotient of two independent random variables, one being a normal distribution in the numerator and the power of a exponential of parameter equal to two at the denominator. The resulting distribution is an extension of the BS distribution that has greater kurtosis values than the usual BS distribution
and the slash Birnbaum-Saunders (SBS) distribution (see [7]). Moments and some properties are derived for the new distribution. Also, we draw inferences by the method of moments and maximum likelihood. A real data application is presented where the model fitting is implemented by using maximum likelihood estimation producing better results than the classic BS model and slash BS model.

Keywords

• Birnbaum-Saunders distribution,Modified slash distribution,Moments,Kurtosis,EM-algorithm

References

1. Akaike, H. A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, 716-723, 1974.
2. Balakrishnan, N., Leiva, V., Sanhueza, A. and Vilca, F. Estimation in the Birnbaum- Saunders distribution based on scale-mixture of normals and the EM-algorithm, Statistics and Operations Research Transactions, 33, 171-192, 2009.
3. Birnbaum, Z.W. and Saunders, S.C. A New Family of Life Distributions, Journal of Applied Probability, 6, 319-327, 1969a.
4. Birnbaum, Z.W. and Saunders, S.C. Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability, 6, 328-347, 1969b.
5. Chen, G. and Balakrishnan, N. A general purpose approximate goodness-of-fit test, Journal of Quality Technology, 27, 154-161, 1995.
6. Díaz-García, J. A. and Leiva, V. A new family of life distributions based on elliptically contoured distributions, Journal of Statistical Planning and Inference, 128, 445-457, 2005.
7. Gómez, H.W.; Olivares-Pacheco, J.F. and Bolfarine, H. An Extension of the Generalized Birnbaun-Saunders Distribution, Statistics and Probability Letters, 79, 331-338, 2009.
8. Ng, H.K.T., Kundu, D. and Balakrishnan, N. Modified moment estimation for the two- parameter Birnbaum-Saunders distribution, Comput. Statist. Data Anal., 43, 283-298, 2003.

9. Lee, S.Y. and Xu, L. Inuence analyses of nonlinear mixed-effects models, Comput. Statist. Data Anal., 45, 321-341, 2004.
10. Leiva, V., Barros, M., Paula, G. A. and Sanhueza, A. Generalized Birnbaum-Saunders distribution applied to air pollutant concentration, Environmetrics, 19, 235-249, 2008.
11. Lin, T.I., Lee, J.C. and Hsieh, W.J. Robust mixture modeling using the skew t distribution, Stat. Comput., 17, 81-92, 2007.
12. Reyes, J., Gómez, H. W. and Bolfarine, H. Modified slash distribution, Statistics, 47, 929- 941, 2013.
13. Rieck, J.R. and Nedelman, J.R. A log-linear model for the Birnbaum-Saunders distribution, Technometrics, 33, 51-60, 1991.
14. Severini, T.A. Likelihood functions for inference in the presence of a nuisance parameter, Biometrika, 85, 507-522, 1998.
15. Vilca, F. and Leiva, V. A new fatigue life model based on family of skew-elliptical distributions, Communications in Statistics: Theory and Methods, 35, 229-244, 2006.