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Proportional Hazard Birnbaum-Saunders Distribution With Application to the Survival Data Analysis
Distribución de riesgo proporcional Birnbaum-Saunders con aplicación al análisis de datos de supervivencia
DOI:
https://doi.org/10.15446/rce.v39n1.55145Keywords:
Birnbaum-Saunders Distribution, Proportional Hazard, Reliability, Survival Data (en)Distribución Birnbaum-Saunders, Riesgo proporcional, Confiabilidad, Datos de supervivencia. (es)
Birnbaum & Saunders (1969b) used a probability distribution to explain the lifetime data and stress produced in materials. Based on this distribution, we propose a generalization of the Birnbaum-Saunders distribution, referred to as the proportional hazard Birnbaum-Saunders distribution, which includes a new parameter that provides more flexibility in terms of skewness and kurtosis than existing models. We derive the main properties of the model. We discuss maximum likelihood estimation of the model parameters. As a natural step, we define the log-linear proportional hazard Birnbaum-Saunders regression model. An empirical application to a real data set is presented in order to illustrate the usefulness of the proposed model. The results showed that the proportional hazard Birnbaum-Saunders model can be used quite effectively in analyzing survival data, reliability problems and
fatigue life studies.
Birnbaum & Saunders (1969b) presentaron una distribución de probabilidad para explicar los datos de supervivencia y estrés producidos sobre los materiales. Basados en esta distribución, proponemos una generalización de la distribución Birnbaum-Saunders, la cual llamamos distribución Birnbaum-Saunders de riesgo proporcional, incluyendo un nuevo parámetro que proporciona una mayor flexibilidad en términos de asimetría y curtosis comparado con los modelos existentes. Derivamos las principales propiedades del modelo. Discutimos la estimación de máxima verosimilitud de los parámetros del modelo. Como un paso natural, definimos el modelo de regresion log-lineal Birnbaum-Saunders de riesgo proporcional. Presentamos una aplicación con un conjunto de datos reales con el propósito de ilustrar la utilidad del modelo propuesto. Los resultados mostraron que el modelo Birnbaum-Saunders de riesgo proporcional puede ser utilizado efectivamente en el análisis de datos de supervivencia, problemas de confiabilidad y estudios de resistencia a la fatiga
1Universidad Industrial de Santander, Facultad de Ciencias, Escuela de Matemáticas, Bucaramanga, Colombia. Associate Professor. Email: gmorenoa@uis.edu.co
2Universidad de Córdoba, Facultad de Ciencias, Departamento de Matemáticas y Estadística, Montería, Colombia. Professor. Email: gmartinez@correo.unicordoba.edu.co
3Instituto Tecnológico Metropolitano, Facultad de Ciencias Exactas y Aplicadas, Medellin, Colombia. Associate Professor. Email: carlosbarrera@itm.edu.co
Birnbaum & Saunders (1969b) used a probability distribution to explain the lifetime data and stress produced in materials. Based on this distribution, we propose a generalization of the Birnbaum-Saunders distribution, referred to as the proportional hazard Birnbaum-Saunders distribution, which includes a new parameter that provides more flexibility in terms of skewness and kurtosis than existing models. We derive the main properties of the model. We discuss maximum likelihood estimation of the model parameters. As a natural step, we define the log-linear proportional hazard Birnbaum-Saunders regression model. An empirical application to a real data set is presented in order to illustrate the usefulness of the proposed model. The results showed that the proportional hazard Birnbaum-Saunders model can be used quite effectively in analyzing survival data, reliability problems and fatigue life studies.
Key words: Birnbaum-Saunders Distribution, Proportional Hazard, Reliability, Survival Data.
Birnbaum & Saunders (1969b) presentaron una distribución de probabilidad para explicar los datos de supervivencia y estrés producidos sobre los materiales. Basados en esta distribución, proponemos una generalización de la distribución Birnbaum-Saunders, la cual llamamos distribución Birnbaum-Saunders de riesgo proporcional, incluyendo un nuevo parámetro que proporciona una mayor flexibilidad en términos de asimetría y curtosis comparado con los modelos existentes. Derivamos las principales propiedades del modelo. Discutimos la estimación de máxima verosimilitud de los parámetros del modelo. Como un paso natural, definimos el modelo de regresion log-lineal Birnbaum-Saunders de riesgo proporcional. Presentamos una aplicación con un conjunto de datos reales con el propósito de ilustrar la utilidad del modelo propuesto. Los resultados mostraron que el modelo Birnbaum-Saunders de riesgo proporcional puede ser utilizado efectivamente en el análisis de datos de supervivencia, problemas de confiabilidad y estudios de resistencia a la fatiga.
Palabras clave: distribución Birnbaum-Saunders, riesgo proporcional, confiabilidad, datos de supervivencia.
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References
1. Akaike, H. (1974), 'A new look at statistical model identification', IEEE Transaction on Automatic Control 19(6), 716-722.
2. Barros, M., Paula, G. A. & Leiva, V. (2008), 'A new class of survival regression models with heavy-tailed errors: robustness and diagnostics', Lifetime Data Analysis 14, 316-332.
3. Birnbaum, Z. & Saunders, S. (1969a), 'Estimation for a family of life distributions with applications to fatigue', Journal of Applied Probability 6, 328-347.
4. Birnbaum, Z. & Saunders, S. (1969b), 'A new family of life distributions', Journal of Applied Probability 6, 319-327.
5. Castillo, E. & Hadi, A. (1995), 'A method for estimating parameters and quantiles of distributions of continuous random variables', Computational Statistics and Data Analysis 20, 421-439.
6. Castillo, N., Gomez, H. & Bolfarine, H. (2011), 'Epsilon birnbaum-saunders distribution family: properties and inference', Statistical Papers 52(2), 871-883.
7. Chan, P., Ng, H., Balakrishnan, N. & Zhou, Q. (2008), 'Point and interval estimation for extreme-value regression model under type-ii censoring', Computational Statistics and Data Analysis 52, 4040-4058.
8. Cisneiros, A., Cribari-Neto, F. & Araújo, C. (2008), 'On birnbaum-saunders inference', Computational Statistics and Data Analysis 52, 4939-4950.
9. Díaz-García, J. & Leiva-Sánchez, V. (2005), 'A new family of life distributions based on the elliptically contoured distributions', Journal of Statistical Planning and Inference 128, 445-457.
10. Efron, B. (1982), The Jackknife, the bootstrap and other resampling plans, CBMS NSF Regional Conference Series in Applied Mathematics.
11. Engelhardt, M., Bain, L. & Wright, F. (1981), 'Inference on the parameters of the Birnbaum-Saunders fatigue life distribution based on maximum likelihood estimation', Americam Statistical Association and Americam Society for Quality 23(3), 251-256.
12. Farias, R., Moreno-Arenas, G. & Patriota, A. (2009), 'Reduction of models in the presence of nuisance parameters', Revista Colombiana de Estadística 32(1), 99-121.
13. From, S. & Li, L. (2006), 'Estimation of the parameters of the birnbaum-saunders distribution', Communications in Statistics: Theory and Methods 35, 2157-2169.
14. Gradshteyn, I. & Ryzhik, I. (2007), Table of Integrals, Series, and Products, Academic Press, New York.
15. Gómez, H., Elal-Olivero, D., Salinas, H. & Bolfarine, H. (2009), 'An extension of the generalized birnbaum-saunders distribution', Statistics and Probability Letters 79, 331-338.
16. Johnson, S., Kotz, S. & Balakrishnan, N. (1995), Continuous Univariate Distributions, Wiley, New York.
17. Leiva, V., Vilca, F., Balakrishnan, N. & Sanhueza, A. (2010), 'A skewed sinh-normal distribution and its properties and application to air pollution', Communications in Statistics: Theory and Methods 39, 426-443.
18. Lemonte, A. (2012), 'A log-birnbaum-saunders regression model with asymmetric errors', Journal of Statistical Computation and Simulation 82, 1775-1787.
19. Lemonte, A., Cribari-Neto, F. & Vasconcellos, K. (2007), 'Improved statistical inference for the two-parameter birnbaum-saunders distribution', Computational Statistics and Data Analysis 51, 4656-4681.
20. Martínez-Flórez, G., Moreno-Arenas, G. & Vergara-Cardozo, S. (2013), 'Properties and inference for proportional hazard models', Revista Colombiana de Estadística 36, 95-114.
21. Nadarajah, S. (2008), 'A truncated inverted beta distribution with application to air pollution data', Stochastic Environmental Research and Risk Assessment 22, 285-289.
22. Ng, H., Kundu, D. & Balakrishnan, N. (2003), 'Modified moment estimation for the two-parameter birnbaum-saunders distribution', Computational Statistics & Data Analysis 43, 283-298.
23. Pewsey, A., Gómez, H. W. & Bolfarine, H. (2012), 'Likelihood based inference for distributions of fractional order statistics', TEST 21, 775-779.
24. Rieck, J. & Nedelman, J. (1991), 'A log-linear model for the birnbaum-saunders distribution', Technometrics 33, 51-60.
25. Santana, L., Vilca, F. & Leiva, V. (2011), 'Influence analysis in skew-birnbaum-saunders regression models and applications', Journal of Applied Statistics 38(8), 1633-1649.
26. Vilca-Labra, F. & Leiva-Sánchez, V. (2006), 'A new fatigue life model based on the family of skew-elliptical distributions', Communications in Statistics-Theory and Methods 35, 229-244.
27. Wu, J. & Wong, A. (2004), 'Improved interval estimation for the two-parameter birnbaum-saunders distribution', Computational Statistics and Data Analysis 47, 809-821.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv39n1a09,
AUTHOR = {Moreno-Arenas, Germán and Martínez-Flórez, Guillermo and Barrera-Causil, Carlos},
TITLE = {{Proportional Hazard Birnbaum-Saunders Distribution With Application to the Survival Data Analysis}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2016},
volume = {39},
number = {1},
pages = {129-147}
}
References
Akaike, H. (1974), ‘A new look at statistical model identification’, IEEE Transaction on Automatic Control 19(6), 716–722.
Barros, M., Paula, G. A. & Leiva, V. (2008), ‘A new class of survival regression models with heavy-tailed errors: robustness and diagnostics’, Lifetime Data Analysis 14, 316–332.
Birnbaum, Z. & Saunders, S. (1969a), ‘Estimation for a family of life distributions with applications to fatigue’, Journal of Applied Probability 6, 328–347.
Birnbaum, Z. & Saunders, S. (1969b), ‘A new family of life distributions’, Journal of Applied Probability 6, 319–327.
Castillo, E. & Hadi, A. (1995), ‘A method for estimating parameters and quantiles of distributions of continuous random variables’, Computational Statistics and Data Analysis 20, 421–439.
Castillo, N., Gomez, H. & Bolfarine, H. (2011), ‘Epsilon birnbaum-saunders distribution family: properties and inference’, Statistical Papers 52(2), 871–883.
Chan, P., Ng, H., Balakrishnan, N. & Zhou, Q. (2008), ‘Point and interval estimation for extreme-value regression model under type-ii censoring’, Computational Statistics and Data Analysis 52, 4040–4058.
Cisneiros, A., Cribari-Neto, F. & Araújo, C. (2008), ‘On birnbaum-saunders inference’, Computational Statistics and Data Analysis 52, 4939–4950.
Díaz-García, J. & Leiva-Sánchez, V. (2005), ‘A new family of life distributions based on the elliptically contoured distributions’, Journal of Statistical Planning and Inference 128, 445–457.
Efron, B. (1982), The Jackknife, the bootstrap and other resampling plans, CBMS NSF Regional Conference Series in Applied Mathematics.
Engelhardt, M., Bain, L. & Wright, F. (1981), ‘Inference on the parameters of the Birnbaum-Saunders fatigue life distribution based on maximum likelihood estimation’, Americam Statistical Association and Americam Society for Quality 23(3), 251–256.
Farias, R., Moreno-Arenas, G. & Patriota, A. (2009), ‘Reduction of models in the presence of nuisance parameters’, Revista Colombiana de Estadística 32(1), 99–121.
From, S. & Li, L. (2006), ‘Estimation of the parameters of the birnbaum-saunders distribution’, Communications in Statistics: Theory and Methods 35, 2157–2169.
Gómez, H., Elal-Olivero, D., Salinas, H. & Bolfarine, H. (2009), ‘An extensión of the generalized birnbaum-saunders distribution’, Statistics and Probability Letters 79, 331–338.
Gradshteyn, I. & Ryzhik, I. (2007), Table of Integrals, Series, and Products, Academic Press, New York.
Johnson, S., Kotz, S. & Balakrishnan, N. (1995), Continuous Univariate Distributions, Wiley, New York.
Leiva, V., Vilca, F., Balakrishnan, N. & Sanhueza, A. (2010), ‘A skewed sinhnormal distribution and its properties and application to air pollution’, Communications in Statistics: Theory and Methods 39, 426–443.
Lemonte, A. (2012), ‘A log-birnbaum-saunders regression model with asymmetric errors’, Journal of Statistical Computation and Simulation 82, 1775–1787.
Lemonte, A., Cribari-Neto, F. & Vasconcellos, K. (2007), ‘Improved statistical inference for the two-parameter birnbaum-saunders distribution’, Computational Statistics and Data Analysis 51, 4656–4681.
Martínez-Flórez, G., Moreno-Arenas, G. & Vergara-Cardozo, S. (2013), ‘Properties and inference for proportional hazard models’, Revista Colombiana de Estadística 36, 95–114.
Nadarajah, S. (2008), ‘A truncated inverted beta distribution with application to air pollution data’, Stochastic Environmental Research and Risk Assessment 22, 285–289.
Ng, H., Kundu, D. & Balakrishnan, N. (2003), ‘Modified moment estimation for the two-parameter birnbaum-saunders distribution’, Computational Statistics & Data Analysis 43, 283–298.
Pewsey, A., Gómez, H. W. & Bolfarine, H. (2012), ‘Likelihood based inference for distributions of fractional order statistics’, TEST 21, 775–779.
Rieck, J. & Nedelman, J. (1991), ‘A log-linear model for the birnbaum-saunders distribution’, Technometrics 33, 51–60.
Santana, L., Vilca, F. & Leiva, V. (2011), ‘Influence analysis in skew-birnbaum-saunders regression models and applications’, Journal of Applied Statistics 38(8), 1633–1649.
Vilca-Labra, F. & Leiva-Sánchez, V. (2006), ‘A new fatigue life model based on the family of skew-elliptical distributions’, Communications in Statistics-Theory and Methods 35, 229–244.
Wu, J. & Wong, A. (2004), ‘Improved interval estimation for the two-parameter birnbaum-saunders distribution’, Computational Statistics and Data Analysis 47, 809–821.
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1. Guillermo Martínez-Flórez, Roger Tovar-Falón, Carlos Barrera-Causil. (2022). Inflated Unit-Birnbaum-Saunders Distribution. Mathematics, 10(4), p.667. https://doi.org/10.3390/math10040667.
2. Guillermo Martínez-Flórez, Sandra Vergara-Cardozo, Roger Tovar-Falón, Luisa Rodriguez-Quevedo. (2023). The Multivariate Skewed Log-Birnbaum–Saunders Distribution and Its Associated Regression Model. Mathematics, 11(5), p.1095. https://doi.org/10.3390/math11051095.
3. Guillermo Martínez-Flórez, Roger Tovar-Falón. (2021). New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications. Mathematics, 9(11), p.1231. https://doi.org/10.3390/math9111231.
4. Guillermo Martínez-Flórez, Heleno Bolfarine, Yolanda M. Gómez. (2021). The Skewed-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model. Symmetry, 13(7), p.1297. https://doi.org/10.3390/sym13071297.
5. Guillermo Martínez-Flórez, David Elal-Olivero, Carlos Barrera-Causil. (2021). Extended Generalized Sinh-Normal Distribution. Mathematics, 9(21), p.2793. https://doi.org/10.3390/math9212793.
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