Research and Innovation Articles
Published 2017-08-09
Keywords
- Birnbaum-Saunders distribution,
- alpha-power distribution,
- power Student t distribution
How to Cite
Moreno-Arenas, G., Martínez-Flórez, G., & Bolfarine, H. (2017). Power Birnbaum-Saunders Student t distribution. Revista Integración, Temas De matemáticas, 35(1), 51–70. https://doi.org/10.18273/revint.v35n1-2017004
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
The fatigue life distribution proposed by Birnbaum and Saunders has been used quite effectively to model times to failure for materials subject to fatigue. In this article, we introduce an extension of the classical Birnbaum-Saunders distribution substituting the normal distribution by the power Student t distribution. The new distribution is more flexible than the classical Birnbaum-Saunders distribution in terms of asymmetry and kurtosis. We discuss maximum likelihood estimation of the model parameters and associated regression model. Two real data set are analysed and the results reveal that the proposed model better some other models proposed in the literature.
MSC2010: 62-07, 62F10, 62J02, 62N86.
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References
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[12] Gupta R.D. and Gupta R.C., “Analyzing skewed data by power normal model”, TEST 17 (2008), No. 1, 197–210.
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[14] Leiva V., Ponce G., Marchant C. and Bustos O., “Fatigue statistical distributions useful for modeling diameter and mortality of trees”, Rev. Colombiana Estadíst. 35 (2012), No. 3, 349–370.
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[16] Lemonte A.J., “A new extension of the Birnbaum-Saunders distribution”, Braz. J. Probab. Stat. 27 (2013), 133–149.
[17] McCool J.I., “Confidence limits for Weibull regression with censored data”, IEEE Trans. on Reliability 29 (1980), 145–150.
[18] Martínez-Flórez G., Bolfarine H. and Gómez H.W., “An alpha-power extension for the Birnbaum-Saunders distribution”, Statistics 48 (2014), No. 4, 896–912.
[19] Martínez-Flórez G., Bolfarine H. and Gómez H.W., “The log-linear Birnbaum-Saunders power model”, Methodol. Comput. Appl. Probab. (2016), 1–21.
[20] Moreno-Arenas G., Martínez-Flórez G. and Barrera-Causil C., “Proportional hazard Birnbaum-Saunders distribution with application to the survival data analysis”, Rev. Colombiana Estadíst. 39 (2016), No. 1, 129–147.
[21] Ng H., Kundu D. and Balakrishnan N., “Modified moment estimation for the two-parameter Birnbaum-Saunders distribution”, Comput. Statist. Data Anal. 43 (2003), No. 3, 283–298.
[22] Paula G., Leiva V., Barros M. and Liu S., “Robust statistical modeling using the Birnbaum-Saunders t distribution applied to insurance”, Appl. Stoch. Models Bus. Ind. 28 (2012), No. 1, 16–34.
[23] R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2015.
[24] Rieck J.R. and Nedelman J.R., “A log-linear model for the Birnbaum-Saunders distribution”, Technometrics 33 (1991), No. 1, 51–60.
[25] Saunders S.C., “A family of random variables closed under reciprocation”, J. Amer. Statist. Assoc. 69 (1974), No. 346, 533–539.
[26] Taylor J. and Verbyla A., “Joint modelling of location and scale parameters of the t distribution”, Stat. Model. 4 (2004), No. 2, 91–112.
[27] Vilca-Labra F. and Leiva-Sánchez V., “A new fatigue life model based on the family of skew-elliptical distributions”, Comm. Statist. Theory Methods 35 (2006), No. 2, 229–244.
[28] Vuong Q., “Likelihood ratio tests for model selection and non-nested hypotheses”, Econometrica 57 (1989), No. 2, 307–333.
[2] Berkane M., Kano Y. and Bentler P.M., “Pseudo maximum likelihood estimation in elliptical theory: effects of misspecification”, Comput. Statist. Data Anal. 18 (1994), No. 2, 255–267.
[3] Birnbaum Z.W. and Saunders S.C., “A new family of life distributions”, J. Appl. Probab. 6 (1969), 319–327.
[4] Castillo E. and Hadi A., “A method for estimating parameters and quantiles of distributions of continuous random variables”, Comput. Statist. Data Anal. 20 (1995), No. 4, 421–439.
[5] Castillo N.O., Gómez H.W. and Bolfarine H., “Epsilon Birnbaum-Saunders distribution family: properties and inference”, Statist. Papers 52 (2011), No. 4, 871–883.
[6] Cordeiro G.M. and Lemonte A.J., “The exponentiated generalized Birnbaum-Saunders distribution”, Appl. Math. Comput. 247 (2014), 762–779.
[7] Chan P.S., Ng H.K.T., Balakrishnan N. and Zhou Q., “Point and interval estimation for extreme-value regression model under Type-II censoring”, Comput. Statist. Data Anal. 52 (2008), No. 8, 4040–4058.
[8] Díaz-García J.A. and Leiva-Sánchez V., “A new family of life distributions based on the elliptically contoured distributions”, J. Statist. Plann. Inference 128 (2005), No. 2, 445–457.
[9] Durrans S.R., “Distributions of fractional order statistics in hydrology”, Water Resources Research 28 (1992), No. 6, 1649–1655.
[10] Fernández C. and Steel M., “Multivariate Student t regression models: Pitfalls and inference”, Biometrika 86 (1999), No. 1, 153–167.
[11] Gómez H., Olivares J. and Bolfarine H., “An extension of the generalized Birnbaum-Saunders distribution”, Statist. Probab. Lett. 79 (2009), 331–338.
[12] Gupta R.D. and Gupta R.C., “Analyzing skewed data by power normal model”, TEST 17 (2008), No. 1, 197–210.
[13] Lange K.L., Little R.J.A. and Taylor J.M.G., “Robust statistical modeling using the t distribution”, J. Amer. statist. Assoc. 84 (1989), No. 408, 881–896.
[14] Leiva V., Ponce G., Marchant C. and Bustos O., “Fatigue statistical distributions useful for modeling diameter and mortality of trees”, Rev. Colombiana Estadíst. 35 (2012), No. 3, 349–370.
[15] Lemonte A.J., “A log-Birnbaum-Saunders regression model with asymmetric errors”, J. Stat. Comput. Simul. 82 (2012), 1775–1787.
[16] Lemonte A.J., “A new extension of the Birnbaum-Saunders distribution”, Braz. J. Probab. Stat. 27 (2013), 133–149.
[17] McCool J.I., “Confidence limits for Weibull regression with censored data”, IEEE Trans. on Reliability 29 (1980), 145–150.
[18] Martínez-Flórez G., Bolfarine H. and Gómez H.W., “An alpha-power extension for the Birnbaum-Saunders distribution”, Statistics 48 (2014), No. 4, 896–912.
[19] Martínez-Flórez G., Bolfarine H. and Gómez H.W., “The log-linear Birnbaum-Saunders power model”, Methodol. Comput. Appl. Probab. (2016), 1–21.
[20] Moreno-Arenas G., Martínez-Flórez G. and Barrera-Causil C., “Proportional hazard Birnbaum-Saunders distribution with application to the survival data analysis”, Rev. Colombiana Estadíst. 39 (2016), No. 1, 129–147.
[21] Ng H., Kundu D. and Balakrishnan N., “Modified moment estimation for the two-parameter Birnbaum-Saunders distribution”, Comput. Statist. Data Anal. 43 (2003), No. 3, 283–298.
[22] Paula G., Leiva V., Barros M. and Liu S., “Robust statistical modeling using the Birnbaum-Saunders t distribution applied to insurance”, Appl. Stoch. Models Bus. Ind. 28 (2012), No. 1, 16–34.
[23] R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2015.
[24] Rieck J.R. and Nedelman J.R., “A log-linear model for the Birnbaum-Saunders distribution”, Technometrics 33 (1991), No. 1, 51–60.
[25] Saunders S.C., “A family of random variables closed under reciprocation”, J. Amer. Statist. Assoc. 69 (1974), No. 346, 533–539.
[26] Taylor J. and Verbyla A., “Joint modelling of location and scale parameters of the t distribution”, Stat. Model. 4 (2004), No. 2, 91–112.
[27] Vilca-Labra F. and Leiva-Sánchez V., “A new fatigue life model based on the family of skew-elliptical distributions”, Comm. Statist. Theory Methods 35 (2006), No. 2, 229–244.
[28] Vuong Q., “Likelihood ratio tests for model selection and non-nested hypotheses”, Econometrica 57 (1989), No. 2, 307–333.