Published

2013-07-01

Inference for the Weibull Distribution Based on Fuzzy Data

Inferencia para la distribución Weibull basada en datos difusos

Keywords:

Bayesian estimation, EM algorithm, Fuzzy data analysis, Maximum likelihood principle. (en)
algoritmo EM, análisis de datos difusos, estimación Bayesiana, principio de máxima verosimilitud (es)

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Authors

  • Abbas Pak Shahid Chamran University of Ahvaz
  • Gholam Ali Parham Shahid Chamran University of Ahvaz
  • Mansour Saraj Shahid Chamran University of Ahvaz

Classical estimation procedures for the parameters of Weibull distribution are based on precise data. It is usually assumed that observed data are precise real numbers. However, some collected data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. In this paper, different methods of estimation are discussed for the parameters of Weibull distribution when the available data are in the form of fuzzy numbers. They include the maximum likelihood  estimation, Bayesian estimation and method of moments. The estimation procedures are discussed in details and compared via Monte Carlo simulations in terms of their average biases and mean squared errors. Finally, a real data set taken from a light emitting diodes manufacturing process is investigated to illustrate the applicability of the proposed methods.

Los procedimientos clásicos de estimación para los parámetros de la distribución Weibull se encuentran basados en datos precisos. Se asume usualmente que los datos observados son números reales precisos. Sin embargo, algunos datos recolectados podrían ser imprecisos y ser representados en la forma de números difusos. Por lo tanto, es necesario generalizar los métodos de estimación estadísticos clásicos de números reales a números difusos. En este artículo, diferentes métodos de estimación son discutidos para los parámetros de la distribución Weibull cuando los datos disponibles están en la forma de números difusos. Estos incluyen la estimación por máxima verosimilitud, la estimación Bayesiana y el método de momentos. Los procedimientos de estimación se discuten en detalle y se comparan vía simulaciones de Monte Carlo en términos de sesgos promedios y errores cuadráticos medios.

Inference for the Weibull Distribution Based on Fuzzy Data

Inferencia para la distribución Weibull basada endatos difusos

ABBAS PAK1, GHOLAM ALI PARHAM2, MANSOUR SARAJ3

1Shahid Chamran University of Ahvaz, Faculty of Mathematical Sciences and Computer, Department of Statistics, Ahvaz, Iran. PhD Student. Email: a-pak@scu.ac.ir
2Shahid Chamran University of Ahvaz, Faculty of Mathematical Sciences and Computer, Department of Statistics, Ahvaz, Iran. Associate professor. Email: parham-g@scu.ac.ir
3Shahid Chamran University of Ahvaz, Faculty of Mathematical Sciences and Computer, Department of Mathematics, Ahvaz, Iran. Associate professor. Email: seraj.a@scu.ac.ir

Abstract

Classical estimation procedures for the parameters of Weibull distribution are based on precise data. It is usually assumed that observed data are precise real numbers. However, some collected data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. In this paper, different methods of estimation are discussed for the parameters of Weibull distribution when the available data are in the form of fuzzy numbers. They include the maximum likelihood estimation, Bayesian estimation and method of moments. The estimation procedures are discussed in details and compared via Monte Carlo simulations in terms of their average biases and mean squared errors. Finally, a real data set taken from a light emitting diodes manufacturing process is investigated to illustrate the applicability of the proposed methods.

Key words: Bayesian estimation, EM algorithm, Fuzzy data analysis, Maximum likelihood principle.

Resumen

Los procedimientos clásicos de estimación para los parámetros de la distribución Weibull se encuentran basados en datos precisos. Se asume usualmente que los datos observados son números reales precisos. Sin embargo, algunos datos recolectados podrían ser imprecisos y ser representados en la forma de números difusos. Por lo tanto, es necesario generalizar los métodos de estimación estadísticos clásicos de números reales a números difusos. En este artículo, diferentes métodos de estimación son discutidos para los parámetros de la distribución Weibull cuando los datos disponibles están en la forma de números difusos. Estos incluyen la estimación por máxima verosimilitud, la estimación Bayesiana y el método de momentos. Los procedimientos de estimación se discuten en detalle y se comparan vía simulaciones de Monte Carlo en términos de sesgos promedios y errores cuadráticos medios.

Palabras clave: algoritmo EM, análisis de datos difusos, estimación Bayesiana, principio de máxima verosimilitud.

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References

1. Ageel, M. I. (2002), 'A novel means of estimating quantiles for 2-parameter Weibull distribution under the right random censoring model', Journal of Computational and Applied Mathematics 149(2), 373-380.

2. Al-Baidhani, P. A. & Sinclair, C. (1987), 'Comparison of methods of estimation of parameters of the Weibull distribution', Communications in Statistics-Simulation and Computation 16(2), 373-384.

3. Balakrishnan, N. & Kateri, M. (2008), 'On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data', Statistics and Probability Letters 78(17), 2971-2975.

4. Balakrishnan, N. & Mitra, D. (2012), 'Left truncated and right censored Weibull data and likelihood inference with an illustration', Computational Statistics and Data Analysis 56, 4011-4025.

5. Banerjee, A. & Kundu, D. (2012), 'Inference based on type-II hybrid censored data from a Weibull distribution', IEEE Transactions on Reliability 57(2), 369-378.

6. Denoeux, T. (2011), 'Maximum likelihood estimation from fuzzy data using the EM algorithm', Fuzzy Sets and Systems 183(1), 72-91.

7. Dubois, D. & Prade, H. (1980), Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York.

8. Gertner, G. Z. & Zhu, H. (1996a), 'Bayesian estimation in forest surveys when samples or prior information are fuzzy', Fuzzy Sets and Systems 77, 277-290.

9. Gertner, G. Z. & Zhu, H. (1996b), 'Bayesian estimation in forest surveys when samples or prior information are fuzzy', Fuzzy Sets and Systems 77, 277-290.

10. Helu, A., Abu-Salih, M. & Alkam, O. (2010), 'Bayes estimation of Weibull distribution parameters using ranked set sampling', Communications in Statistics- Theory and Methods 39(14), 2533-2551.

11. Joarder, A., Krishna, H. & Kundu, D. (2011), 'Inferences on Weibull parameters with conventional type-I censoring', Computational Statistics and Data Analysis 55, 1-11.

12. Lin, C., Chou, C. & Huang, Y. (2012), 'Inference for the Weibull distribution with progressive hybrid censoring', Computational Statistics and Data Analysis 56, 451-467.

13. Marks, N. B. (2005), 'Estimation of Weibull parameters from common percentiles', Journal of Applied Statistics 32(1), 17-24.

14. Nandi, S. & Dewan, I. (2010), 'An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring', Computational Statistics and Data Analysis 54(6), 1559-1569.

15. Ng, H. K. T. & Wang, Z. (2009), 'Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval censored sample', Journal of Statistical Computation and Simulation 79(2), 145-159.

16. Press, S. J. (2001), The Subjectivity of Scientists and the Bayesian Approach, Wiley, New York.

17. Qiao, O. & Tsokos, C. P. (1994), 'Parameter estimation of the Weibull probability distribution', Mathematics and Computers in Simulation 37, 47-55.

18. R Development Core Team, (2011), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *http://www.R-project.org

19. Singpurwalla, N. D. & Booker, J. M. (2004), 'Membership functions and probability measures of fuzzy sets', Journal of the American Statistical Association 99(467), 867-877.

20. Tan, Z. (2009), 'A new approach to MLE of Weibull distribution with interval data', Reliability Engineering and System Safety 94(2), 394-403.

21. Tierney, L. & Kadane, J. B. (1986), 'Accurate approximations for posterior moments and marginal densities', Journal of the American Statistical Association 81, 82-86.

22. Watkins, A. J. (1994), 'On maximum likelihood estimation for the two parameter Weibull distribution', Microelectronics Reliability 36(5), 595-603.

23. Zadeh, L. A. (1968), 'Probability measures of fuzzy events', Journal of Mathematical Analysis and Applications 10, 421-427.

[Recibido en febrero de 2013. Aceptado en septiembre de 2013]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv36n2a10,
    AUTHOR  = {Pak, Abbas and Parham, Gholam Ali and Saraj, Mansour},
    TITLE   = {{Inference for the Weibull Distribution Based on Fuzzy Data}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {2},
    pages   = {337-356}
}

References

Ageel, M. I. (2002), ‘A novel means of estimating quantiles for 2-parameter Weibull distribution under the right random censoring model’, Journal of Computational and Applied Mathematics 149(2), 373–380.

Al-Baidhani, P. A. & Sinclair, C. (1987), ‘Comparison of methods of estimation of parameters of the Weibull distribution’, Communications in Statistics- Simulation and Computation 16(2), 373–384.

Balakrishnan, N. & Kateri, M. (2008), ‘On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data’, Statistics and Probability Letters 78(17), 2971–2975.

Balakrishnan, N. & Mitra, D. (2012), ‘Left truncated and right censored Weibull data and likelihood inference with an illustration’, Computational Statistics and Data Analysis 56, 4011–4025.

Banerjee, A. & Kundu, D. (2012), ‘Inference based on type-II hybrid censored data from a Weibull distribution’, IEEE Transactions on Reliability 57(2), 369–378.

Denoeux, T. (2011), ‘Maximum likelihood estimation from fuzzy data using the EM algorithm’, Fuzzy Sets and Systems 183(1), 72–91.

Dubois, D. & Prade, H. (1980), Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York.

Gertner, G. Z. & Zhu, H. (1996), ‘Bayesian estimation in forest surveys when samples or prior information are fuzzy’, Fuzzy Sets and Systems 77, 277–290.

Helu, A., Abu-Salih, M. & Alkam, O. (2010), ‘Bayes estimation of Weibull distribution parameters using ranked set sampling’, Communications in Statistics-Theory and Methods 39(14), 2533–2551.

Joarder, A., Krishna, H. & Kundu, D. (2011), ‘Inferences on Weibull parameters with conventional type-I censoring’, Computational Statistics and Data Analysis 55, 1–11.

Lin, C., Chou, C. & Huang, Y. (2012), ‘Inference for the Weibull distribution with progressive hybrid censoring’, Computational Statistics and Data Analysis 56, 451–467.

Marks, N. B. (2005), ‘Estimation ofWeibull parameters from common percentiles’, Journal of Applied Statistics 32(1), 17–24.

Nandi, S. & Dewan, I. (2010), ‘An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring’, Computational Statistics and Data Analysis 54(6), 1559–1569.

Ng, H. K. T. & Wang, Z. (2009), ‘Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval censored sample’, Journal of Statistical Computation and Simulation 79(2), 145–159.

Press, S. J. (2001), The Subjectivity of Scientists and the Bayesian Approach, Wiley, New York.

Qiao, O. & Tsokos, C. P. (1994), ‘Parameter estimation of the Weibull probability distribution’, Mathematics and Computers in Simulation 37, 47–55.

R Development Core Team (2011), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.

*http://www.R-project.org

Singpurwalla, N. D. & Booker, J. M. (2004), ‘Membership functions and probability measures of fuzzy sets’, Journal of the American Statistical Association 99(467), 867–877.

Tan, Z. (2009), ‘A new approach to MLE of Weibull distribution with interval data’, Reliability Engineering and System Safety 94(2), 394–403.

Tierney, L. & Kadane, J. B. (1986), ‘Accurate approximations for posterior moments and marginal densities’, Journal of the American Statistical Association 81, 82–86.

Watkins, A. J. (1994), ‘On maximum likelihood estimation for the two parameter Weibull distribution’, Microelectronics Reliability 36(5), 595–603.

Zadeh, L. A. (1968), ‘Probability measures of fuzzy events’, Journal of Mathematical Analysis and Applications 10, 421–427.

How to Cite

APA

Pak, A., Parham, G. A. and Saraj, M. (2013). Inference for the Weibull Distribution Based on Fuzzy Data. Revista Colombiana de Estadística, 36(2), 337–356. https://revistas.unal.edu.co/index.php/estad/article/view/44352

ACM

[1]
Pak, A., Parham, G.A. and Saraj, M. 2013. Inference for the Weibull Distribution Based on Fuzzy Data. Revista Colombiana de Estadística. 36, 2 (Jul. 2013), 337–356.

ACS

(1)
Pak, A.; Parham, G. A.; Saraj, M. Inference for the Weibull Distribution Based on Fuzzy Data. Rev. colomb. estad. 2013, 36, 337-356.

ABNT

PAK, A.; PARHAM, G. A.; SARAJ, M. Inference for the Weibull Distribution Based on Fuzzy Data. Revista Colombiana de Estadística, [S. l.], v. 36, n. 2, p. 337–356, 2013. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44352. Acesso em: 28 mar. 2025.

Chicago

Pak, Abbas, Gholam Ali Parham, and Mansour Saraj. 2013. “Inference for the Weibull Distribution Based on Fuzzy Data”. Revista Colombiana De Estadística 36 (2):337-56. https://revistas.unal.edu.co/index.php/estad/article/view/44352.

Harvard

Pak, A., Parham, G. A. and Saraj, M. (2013) “Inference for the Weibull Distribution Based on Fuzzy Data”, Revista Colombiana de Estadística, 36(2), pp. 337–356. Available at: https://revistas.unal.edu.co/index.php/estad/article/view/44352 (Accessed: 28 March 2025).

IEEE

[1]
A. Pak, G. A. Parham, and M. Saraj, “Inference for the Weibull Distribution Based on Fuzzy Data”, Rev. colomb. estad., vol. 36, no. 2, pp. 337–356, Jul. 2013.

MLA

Pak, A., G. A. Parham, and M. Saraj. “Inference for the Weibull Distribution Based on Fuzzy Data”. Revista Colombiana de Estadística, vol. 36, no. 2, July 2013, pp. 337-56, https://revistas.unal.edu.co/index.php/estad/article/view/44352.

Turabian

Pak, Abbas, Gholam Ali Parham, and Mansour Saraj. “Inference for the Weibull Distribution Based on Fuzzy Data”. Revista Colombiana de Estadística 36, no. 2 (July 1, 2013): 337–356. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44352.

Vancouver

1.
Pak A, Parham GA, Saraj M. Inference for the Weibull Distribution Based on Fuzzy Data. Rev. colomb. estad. [Internet]. 2013 Jul. 1 [cited 2025 Mar. 28];36(2):337-56. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44352

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