1Universidad de Valpara\'{\i}so, Departamento de Estad\'{i}stica, Valpara\'{\i}so, Chile. Profesor titular. Email: victor.leiva@uv.cl
2Instituto Nacional de Estadísticas, Santiago, Chile. Ingeniero estadístico. Email: gerson.soto@ine.cl
3Universidad de Valpara\'{\i}so, Departamento de Estad\'{i}stica, Valpara\'{\i}so, Chile. Pontificia Universidad Católica de Valpara\'{\i}so, Instituto de Estad\'{i}stica, Valpara\'{\i}so, Chile. Profesor adjunto. Email: enrique.cabrera@uv.cl
]]>
l modelo Birnbaum-Saunders (BS) es una distribución de vida que tiene propiedades interesantes y aplicaciones en varias áreas. Esto la ha convertido en un foco de investigación importante en el último tiempo. Sin embargo, la suma de variables aleatorias independientes BS (BSsum) no sigue una distribución BS. A través de la distribución BSsum, se pueden monitorear los tiempos de vida de productos expuestos a fallas mediante una carta de control de calidad. Los procedimientos clásicos de cartas de control suponen normalidad en la distribución de los datos. No obstante, una de las características principales de los tiempos de vida es que éstos generalmente siguen distribuciones asimétricas. Por tanto, si se quiere monitorear estos tiempos, se deben considerar cartas de control para distribuciones asimétricas, como es el caso de la distribución BS. El monitoreo de los tiempos de vida se realiza generalmente mediante el tiempo acumulado o el tiempo promedio hasta la ocurrencia de cierto número de fallas. Entonces, usando la distribución BSsum, desarrollamos, implementamos y aplicamos una nueva metodología para cartas de control basada en la distribución BS.
Palabras clave: lenguaje de computación R, métodos de verosimilitud, tiempos de vida.
The Birnbaum-Saunders (BS) model is a life distribution with interesting properties and applications in several fields. This has transformed the BS model in an important research focus in recent decades. However, the sum of BS (BSsum) independent random variables does not follow a BS distribution. By means of the BSsum distribution, we can monitor the lifetime of products subject to failures using a quality control chart. Classic procedures for control charts assume normality in the distribution of the data. Nevertheless, one of the main characteristics of the lifetimes is that them generally follow asymmetric distributions. Therefore, if we want to monitor these lifetimes, we must consider control charts for asymmetric distributions, such as it is the case of the BS distribution. The monitoring of the lifetimes is carried out generally by the accumulated lifetime or the lifetime average until than a number of failures occurs. Thus, by using the BSsum distribution, we develop, implement and apply a new methodology for control charts based on the BS distribution.
Key words: Lifetime data, Likelihood methods, R computer language.
Referencias
1. Balakrishnan, N., Leiva, V. & L\'opez, J. (2007), `Acceptance sampling plans from truncated life tests based on the generalized Birnbaum-Saunders distribution´, Communications in Statistics: Simulation and Computation 36, 643-656. [ Links ]
2. Balakrishnan, N., Leiva, V., Sanhueza, A. & Cabrera, E. (2009), `Mixture inverse Gaussian distribution and its transformations, moments and applications´, Statistics 43, 91-104. [ Links ]
3. 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. [ Links ]
4. Barros, M., Paula, G. A. & Leiva, V. (2009), `An R implementation for generalized Birnbaum-Saunders distributions´, Computational Statistics and Data Analysis 53, 1511-1528. [ Links ]
5. Bhatti, C. R. (2010), `The Birnbaum-Saunders autoregressive conditional duration model´, Mathematics and Computers in Simulation 80, 2062-2078. [ Links ]
6. Birnbaum, Z. W. & Saunders, S. C. (1969a), `A new family of life distributions´, Journal of Applied Probability 6, 319-327. [ Links ]
7. Birnbaum, Z. W. & Saunders, S. C. (1969b), `Estimation for a family of life distributions with applications to fatigue´, Journal of Applied Probability 6, 328-347. [ Links ]
8. Chang, D. S. & Tang, L. C. (1994), `Graphical analysis for Birbaum-Saunders distribution´, Microelectronics and Reliability 34, 17-22. [ Links ]
9. Cheng, S. W. & Xie, H. (2000), `Control charts for lognormal data´, Tamkang Journal of Science and Engineering 3, 131-137. [ Links ]
10. D\'iaz-Garc\'\ia, J. A. & Leiva, V. (2005), `A new family of life distributions based on elliptically contoured distributions´, Journal of Statistical Planning and Inference 128, 445-457. [ Links ]
11. Desmond, A. (1985), `Stochastic models of failure in random environments´, Canadian Journal of Statistics 13, 171-183. [ Links ]
12. Duncan, A. (1996), Control de Calidad y Estad\'\istica Industrial, Alfaomega Grupo Editor, M\'exico. [ Links ]
13. Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995), Continuous Univariate Distributions, Vol. 2, Second edn, John Wiley & Sons, New York. [ Links ]
14. Lawless, J. F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York. [ Links ]
15. Leiva, V., Barros, M., Paula, G. A. & Galea, M. (2007), `Influence diagnostics in log-Birnbaum-Saunders regression models with censored data´, Computational Statistics and Data Analysis 51, 5694-5707. [ Links ]
16. Leiva, V., Barros, M., Paula, G. & Sanhueza, D. (2008), `Generalized Birnbaum-Saunders distributions applied to air pollutant concentration´, Environmetrics 19, 235-249. [ Links ]
17. Leiva, V., Hern\'andez, H. & Riquelme, M. (2006), `A new package for the birnbaum-saunders distribution´, R Journal 6, 35-40. *http://www.R-project.org/doc/Rnews/Rnews_2006-4.pdf [ Links ]
18. Leiva, V., Sanhueza, A. & Angulo, J. M. (2009), `A length-biased version of the Birnbaum-Saunders distribution with application in water quality´, Stochastic Environmental Research and Risk Assessment 23, 299-307. [ Links ]
19. Leiva, V., Sanhueza, A. & Saunders, S. C. (2009), New developments and applications on life distributions under cumulative damage, 4, CIMFAV. *http://www.cimfav.cl/reports.html#2009 [ Links ]
20. Leiva, V., Sanhueza, A., Sen, P. K. & Paula, G. A. (2008), `Random number generators for the generalized Birnbaum-Saunders distribution´, Journal of Statistical Computation and Simulation 78, 1105-1118. [ Links ]
21. 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. [ Links ]
22. Lio, Y. L. & Park, C. (2008), `A bootstrap control chart for Birnbaum-Saunders percentiles´, Quality and Reliability Engineering International 24, 85-600. [ Links ]
23. Marshall, A. W. & Olkin, I. (2007), Life Distributions, Springer, New York. [ Links ]
24. Ng, H. K. T., Kundu, D. & Balakrishnan, N. (2003), `Modified moment estimation for the two-parameter Birnbaum-Saunders distribution´, Computational Statistics and Data Analysis 43, 283-298. [ Links ]
25. P\'erez, R. A. & Correa, J. C. (2008), `Intervalos de confianza v\'\ia verosimilitud relativa de los par\'ametros de la distribuci\'on Birnbaum-Saunders´, Revista Colombiana de Estad\'\istica 37, 645-670. [ Links ]
26. Podlaski, R. (2008), `Characterization of diameter distribution data in near-natural forests using the Birnbaum-Saunders distribution´, Canadian Journal of Forest Research 18, 518-526. [ Links ]
27. R Development Core Team, (2009), R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. *http://www.R-project.org [ Links ]
28. Raaijmakers, F. J. M. (1980), `The lifetime of a standby system of units having the Birnbaum-Saunders distribution´, Journal of Applied Probability 17, 490-497. [ Links ]
29. Raaijmakers, F. J. M. (1981), `Reliability of standby system for units with the Birnbaum-Saunders distribution´, IEEE Transactions on Reliability 30, 198-199. [ Links ]
30. Sanhueza, A., Leiva, V. & Balakrishnan, N. (2008), `The generalized Birnbaum-Saunders distribution and its theory, methodology and application´, Communications in Statistics: Theory and Methods 37, 645-670. [ Links ]
31. Saunders, S. C. (2007), Reliability Life Testing and Prediction of Service Lives, Springer, New York. [ Links ]
32. Schoonhoven, M. & Does, R. J. M. M. (2010), `The control chart under non-normality´, Quality and Reliability Engineering International 26, 167-176. [ Links ]
33. Steiner, H. S. & Mackay, J. R. (2000), `Monitoring processes with highly censored data´, Journal of Quality Technology 32, 199-208. [ Links ]
34. Surucu, B. & Sazak, H. (2009), `Monitoring reliability for a three-parameter Weibull distribution´, Reliability Engineenring and System Safety 94, 503-508. [ Links ]
35. Vargas, J. & Monta\~no, T. (2005), `Carta de control CEV \overline{X} para distribuciones Weibull con datos censurados´, Revista Colombiana de Estad\'\istica 28, 125-139. [ Links ]
36. Vilca, F., Sanhueza, A., Leiva, V. & Christakos, G. (2010), `An extended Birnbaum-Saunders model and its application in the study of environmental quality in Santiago, Chile´, Stochastic Environmental Research and Risk Assessment 24, 771-782. [ Links ]
37. Zhang, L. & Chen, G. (2004), `EWMA charts for monitoring the mean of censored Weibull lifetimes´, Journal of Quality Technology 36, 321-328. [ Links ]
@ARTICLE{RCEv34n1a08,
AUTHOR = {Leiva, Víctor and Soto, Gerson and Cabrera, Enrique and Cabrera, Guillermo},
TITLE = {{Nuevas cartas de control basadas en la distribución Birnbaum-Saunders y su implementación}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2011},
volume = {34},
number = {1},
pages = {147-176}
}