Estimation Based on Progressive First-Failure Censored Sampling with Binomial Removals

Affiliation(s)

Faculty of Science, Islamic University, Madinah, Saudi Arabia.

Mathematics Department, Sohag University, Sohag, Egypt.

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.

Faculty of Science, Islamic University, Madinah, Saudi Arabia.

Mathematics Department, Sohag University, Sohag, Egypt.

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.

ABSTRACT

In this paper, the inference for the Burr-X model under
progressively first-failure censoring scheme is discussed. Based on this new
censoring were the number of units removed at each failure time has a discrete
binomial distribution. The maximum likelihood, Bootstrap and Bayes estimates
for the Burr-X distribution are obtained. The Bayes estimators are obtained
using both the symmetric and asymmetric loss functions. Approximate confidence
interval and highest posterior density interval (HPDI) are discussed. A
numerical example is provided to illustrate the proposed estimation methods
developed here. The maximum likelihood and the different Bayes estimates are
compared via a Monte Carlo simulation study.

Cite this paper

Soliman, A. , Ellah, A. , Abou-Elheggag, N. and El-Sagheer, R. (2013) Estimation Based on Progressive First-Failure Censored Sampling with Binomial Removals.*Intelligent Information Management*, **5**, 117-125. doi: 10.4236/iim.2013.54012.

Soliman, A. , Ellah, A. , Abou-Elheggag, N. and El-Sagheer, R. (2013) Estimation Based on Progressive First-Failure Censored Sampling with Binomial Removals.

References

[1] L. G. Johnson, “Theory and Technique of Variation Research,” Elsevier, Amsterdam, 1964.

[2] S.-J. Wu and C. Kus, “On Estimation Based on Progressive First-Failure-Censored Sampling,” Computational Statistics and Data Analysis, Vol. 53, No. 10, 2009, pp. 3659-3670. doi:10.1016/j.csda.2009.03.010

[3] P. L. Gupta, S. Gupta and Ya. Lvin, “Analysis of Failure Time Data by Burr Distribution,” Communication Statistics Theory & Methods, Vol. 25, No. 9, 1996, pp. 2013-2024. doi:10.1080/03610929608831817

[4] A. Childs and N. Balakrishnan, “Conditional Inference Procedures for the Laplace Distribution When the Observed Samples Are Progressively Censored,” Metrika, Vol. 52, No. 3, 2000, pp. 253-265. doi:10.1007/s001840000092

[5] S. K. Tse, C. Y. Yang and H.-K. Yuen, “Statistical Analysis of Weibull Distribution Lifetime Data under Type II Progressive Censoring with Binomial Removals,” Journal of Applied Statistics, Vol. 27, No. 8, 2000, pp. 1033-1043. doi:10.1080/02664760050173355

[6] M. A. M. Ali Mousa and Z. F. Jaheen, “Statistical Inference for the Burr Model Based on Progressively Censored Data,” An International Computers & Mathematics with Applications, Vol. 43, No. 10-11, 2002, pp. 1441-1449. doi:10.1016/S0898-1221(02)00110-4

[7] K. Ng, P. S. Chan and N. Balakrishnan, “Estimation of Parameters from Progressively Censored Data Using an Algorithm,” Computational Statistics and Data Analysis, Vol. 39, No. 4, 2002, pp. 371-386. doi:10.1016/S0167-9473(01)00091-3

[8] S.-J. Wu and C.-T. Chang, “Parameter Estimation Based on Exponential Progressive Type II Censored Data with Binomial Removals,” Information and Mangement Statistics, Vol. 13, No. 3, 2002, pp. 37-46.

[9] N. Balakrishnan, N. Knnan, C. T. Lin and H. Ng, “Point and Interval Estimation for Gaussian Distribution Based on Progressively Type-II Censored Samples,” IEEE Transactions on Reliability, Vol. 52, No. 3, 2003, pp. 90-95. doi:10.1109/TR.2002.805786

[10] S.-J. Wu, “Estimation for the Two-Parameter Pareto Distribution under Progressive Censoring with Uniform Removals,” Journal of Statistical Computation and Simulation, Vol. 73, No. 2, 2003, pp. 125-134. doi:10.1080/00949650215732

[11] A. A. Soliman, “Estimation of Parameters of Life from Progressively Censored Data Using Burr-XII Model,” IEEE Transactions on Reliability, Vol. 54, No. 1, 2005, pp. 34-42. doi:10.1109/TR.2004.842528

[12] A. M. Sarhan and A. Abuammoh, “Statistical Inference Using Progressively Type-II Censored Data with Random Scheme,” International Mathematical Forum, Vol. 35, No. 3, 2008, pp. 1713-1725.

[13] B. Efron, “The Bootstrap and Other Resampling Plans,” CBMS-NSF Regional Conference Seriesin Applied Mathematics, Vol. 38, SIAM, Philadelphia, 1982.

[14] N. Balakrishnan and R. Asandhu, “A Simple Simulation Algorithm for Generating Progressively Type-II Censored Samples,” The American Statistican, Vol. 49, No. 2, 1995, pp. 229-230.

[15] A. P. Basu and N. Ebrahimi, “Bayesian Approach to Life Testing and Reliability Estimation Using Asymmetric Loss Function,” Journal of Statistical Planning and Inference, Vol. 29, No. 1-2, 1991, pp. 21-31. doi:10.1016/0378-3758(92)90118-C

[16] R. V. Caneld, “A Bayesian Approach to Reliability Estimation Using a Loss Function,” IEEE Transactions on Reliability, Vol. 19, No. 1, 1970, pp. 13-16.

[17] W. J. Zimmer, J. Bert Keats and F. K. Wang, “The Burr XII Distribution in Reliability Analysis,” Journal of Quality & Technology, Vol. 30, No. 4, 1998, pp. 386-394.

[18] U. Balasooriya and N. Balakrishnan, “Reliability Sampling Plans for Log-Normal Distribution Based on Progressively Censored Samples,” IEEE Transactions on Reliability, Vol. 49, No. 2, 2000, pp. 199-203. doi:10.1109/24.877338

[19] A. A. Soliman, “Comparison of LINEX and Quadratic Bayes Estimators for the Rayleigh Distribution,” Communications in Statistics Theory and Methods, Vol. 29, No. 1, 2000, pp. 95-107. doi:10.1080/03610920008832471

[20] A. A. Soliman, “Estimation of Parameters of Life from Progressively Censored Data Using Burr-XII Model,” IEEE Transactions on Reliability, Vol. 54, No. 1, 2005, pp. 34-42.

[21] D. K. Dey, M. Ghosh and C. Srinivasan, “Simultaneous Estimation of Parameters under Entropy Loss,” Journal of Statistical Planning and Inference, Vol. 15, No. 2, 1987, pp. 347-363.

[22] D. K. Dey and P. L. Liu, “On Comparison of Estimation in Generalized Life Model,” Microelectron & Reliability, Vol. 32, No. 1-2, 1992, pp. 207-221. doi:10.1016/0026-2714(92)90099-7

[1] L. G. Johnson, “Theory and Technique of Variation Research,” Elsevier, Amsterdam, 1964.

[2] S.-J. Wu and C. Kus, “On Estimation Based on Progressive First-Failure-Censored Sampling,” Computational Statistics and Data Analysis, Vol. 53, No. 10, 2009, pp. 3659-3670. doi:10.1016/j.csda.2009.03.010

[3] P. L. Gupta, S. Gupta and Ya. Lvin, “Analysis of Failure Time Data by Burr Distribution,” Communication Statistics Theory & Methods, Vol. 25, No. 9, 1996, pp. 2013-2024. doi:10.1080/03610929608831817

[4] A. Childs and N. Balakrishnan, “Conditional Inference Procedures for the Laplace Distribution When the Observed Samples Are Progressively Censored,” Metrika, Vol. 52, No. 3, 2000, pp. 253-265. doi:10.1007/s001840000092

[5] S. K. Tse, C. Y. Yang and H.-K. Yuen, “Statistical Analysis of Weibull Distribution Lifetime Data under Type II Progressive Censoring with Binomial Removals,” Journal of Applied Statistics, Vol. 27, No. 8, 2000, pp. 1033-1043. doi:10.1080/02664760050173355

[6] M. A. M. Ali Mousa and Z. F. Jaheen, “Statistical Inference for the Burr Model Based on Progressively Censored Data,” An International Computers & Mathematics with Applications, Vol. 43, No. 10-11, 2002, pp. 1441-1449. doi:10.1016/S0898-1221(02)00110-4

[7] K. Ng, P. S. Chan and N. Balakrishnan, “Estimation of Parameters from Progressively Censored Data Using an Algorithm,” Computational Statistics and Data Analysis, Vol. 39, No. 4, 2002, pp. 371-386. doi:10.1016/S0167-9473(01)00091-3

[8] S.-J. Wu and C.-T. Chang, “Parameter Estimation Based on Exponential Progressive Type II Censored Data with Binomial Removals,” Information and Mangement Statistics, Vol. 13, No. 3, 2002, pp. 37-46.

[9] N. Balakrishnan, N. Knnan, C. T. Lin and H. Ng, “Point and Interval Estimation for Gaussian Distribution Based on Progressively Type-II Censored Samples,” IEEE Transactions on Reliability, Vol. 52, No. 3, 2003, pp. 90-95. doi:10.1109/TR.2002.805786

[10] S.-J. Wu, “Estimation for the Two-Parameter Pareto Distribution under Progressive Censoring with Uniform Removals,” Journal of Statistical Computation and Simulation, Vol. 73, No. 2, 2003, pp. 125-134. doi:10.1080/00949650215732

[11] A. A. Soliman, “Estimation of Parameters of Life from Progressively Censored Data Using Burr-XII Model,” IEEE Transactions on Reliability, Vol. 54, No. 1, 2005, pp. 34-42. doi:10.1109/TR.2004.842528

[12] A. M. Sarhan and A. Abuammoh, “Statistical Inference Using Progressively Type-II Censored Data with Random Scheme,” International Mathematical Forum, Vol. 35, No. 3, 2008, pp. 1713-1725.

[13] B. Efron, “The Bootstrap and Other Resampling Plans,” CBMS-NSF Regional Conference Seriesin Applied Mathematics, Vol. 38, SIAM, Philadelphia, 1982.

[14] N. Balakrishnan and R. Asandhu, “A Simple Simulation Algorithm for Generating Progressively Type-II Censored Samples,” The American Statistican, Vol. 49, No. 2, 1995, pp. 229-230.

[15] A. P. Basu and N. Ebrahimi, “Bayesian Approach to Life Testing and Reliability Estimation Using Asymmetric Loss Function,” Journal of Statistical Planning and Inference, Vol. 29, No. 1-2, 1991, pp. 21-31. doi:10.1016/0378-3758(92)90118-C

[16] R. V. Caneld, “A Bayesian Approach to Reliability Estimation Using a Loss Function,” IEEE Transactions on Reliability, Vol. 19, No. 1, 1970, pp. 13-16.

[17] W. J. Zimmer, J. Bert Keats and F. K. Wang, “The Burr XII Distribution in Reliability Analysis,” Journal of Quality & Technology, Vol. 30, No. 4, 1998, pp. 386-394.

[18] U. Balasooriya and N. Balakrishnan, “Reliability Sampling Plans for Log-Normal Distribution Based on Progressively Censored Samples,” IEEE Transactions on Reliability, Vol. 49, No. 2, 2000, pp. 199-203. doi:10.1109/24.877338

[19] A. A. Soliman, “Comparison of LINEX and Quadratic Bayes Estimators for the Rayleigh Distribution,” Communications in Statistics Theory and Methods, Vol. 29, No. 1, 2000, pp. 95-107. doi:10.1080/03610920008832471

[20] A. A. Soliman, “Estimation of Parameters of Life from Progressively Censored Data Using Burr-XII Model,” IEEE Transactions on Reliability, Vol. 54, No. 1, 2005, pp. 34-42.

[21] D. K. Dey, M. Ghosh and C. Srinivasan, “Simultaneous Estimation of Parameters under Entropy Loss,” Journal of Statistical Planning and Inference, Vol. 15, No. 2, 1987, pp. 347-363.

[22] D. K. Dey and P. L. Liu, “On Comparison of Estimation in Generalized Life Model,” Microelectron & Reliability, Vol. 32, No. 1-2, 1992, pp. 207-221. doi:10.1016/0026-2714(92)90099-7