Planning Life Tests for Burr XII Distributed Products Under Progressive Group-censoring with Cost Considerations

Yıl 2012, Cilt: 25 Sayı: 2, 425 - 434, 03.01.2012

Coşkun Kus , Yunus Akdogan , Shuo-jye Wu

Öz

In this paper, a progressive type I group-censoring life test for Burr XII distribution is considered. We use the maximum likelihood method to obtain the point estimators of the Burr XII parameters. The approximate confidence intervals for the parameters of Burr XII distribution are also obtained. We use modified algorithm proposed by Kus et al. [19] to decide the number of test units, number of inspections, and length of inspection interval under a restricted budget of experiment such that the determinant of the asymptotic variances-covariance of estimators of parameters is minimum. A numerical example is presented to illustrate the proposed approach. The sensitivity analysis is also investigated.

Anahtar Kelimeler

D-Optimality, Grouped data, Interval censoring, Nonlinear mixed integer programming, Maximum likelihood method, Progressive censoring.

Kaynakça

• Aggarwala, R. “Progressive interval censoring: some mathematical results with applications to inference” Communications in Statistics - Theory and Methods, 30, 1921-1935 (2001).
• Akdoğan, Y., “Progressively group censoring and optimal design of experiment”, Selçuk University Science Institute published Master Thesis, Konya (2011).
• Akdoğan, Y., Kuş, C. “Kısıtlı Maliyet Durumunda Burr Tip III Dağılımı için Optimal İlerleyen Tür Grup Sansürleme Planı”, Uluslararası 7. İstatistik Kongresi, Antalya, 28 Nisan-01 Mayıs 2011., 4(1): 210-211 (2011).
• Akdoğan, Y., Kuş, C., Wu, S.-J. “Optimal progressive group-censoring plans for Logistic distribution under cost constraint “, 5th Annual International Statistics & Mathematical Education, Athens, Greece, 13-16 June 2011, MAT2011/3379 (2011). on Mathematics,
• AL-Hussaini, E.K., Jaheen, Z.F., Bayesian prediction bounds for the Burr type XII failure model, Communications in Statistics - Theory and Methods, 24 (7): 1829–1842 (1995).
• AL-Hussaini, E.K., Jaheen, Z.F “Bayesian prediction bounds for the Burr type XII distribution in the presence of outliers”, J. Statist. Planning Inference, 23–37, 55 (1996).
• Ali Mousa, M.A.M., Jaheen, Z.F., “Bayesian prediction for the Burr type XII model based on doubly censored data”, Statistics 29, 285–294 (1997).
• Ali Mousa, M.A.M., Jaheen, Z.F., “Bayesian prediction for the two parameter Burr type XII model based on doubly censored data”, J. Appl. Statist. Sci. 7 (2/3) ,103–111 (1998).
• Ali Mousa, M.A.M., Jaheen, Z.F., “Statistical inference progressively censored data”, Comput. Math. Appl. 43, 1441–1449 (2002). model based on
• Ali Mousa, M.A.M., Jaheen, Z.F., “Bayesian prediction for progressively censored data form the Burr model”, Statist. Papers 43, 587–593 (2002).
• Burr, I. W. Cumulative frequency functions. Annals of Mathematical Statistics, 13, 215–232 (1942).
• Burr, I.W., Cislak, P.J., “On a general system of distributions: I. Its curve-shape characteristics: II. The sample median”, J. Am. Statist. Assoc. 63, 627–635 (1968).
• Casella, G., Berger, R. L. “Statistical Inference”, 2nd edition, Duxbury, Pacafic Grove, CA.
• Chen, Z. “A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function”, Statistics and Probability Letters, 49, 155-161 (2000).
• Chen, Z., Mi, J. “Confidence interval for the mean of the exponential distribution, Based on grouped data”. IEEE Transactions on Reliability, 45, 671-677 (1996).
• Cheng, K. F., Chen, C. H. “Estimation of the Weibull parameters with grouped data”. Communications in Statistics - Theory and Methods, 17,325-341 (1988).
• Gupta, P.L., Gupta, R.C., Lvin, S.Ya., “Analysis of failure Communications in Statistics - Theory and Burr
• distribution”, Methods, 25 (9) 2013–2024 (1996).
• Kuş, C., Akdoğan, Y. “Kısıtlı Maliyet Durumunda Bathtube-Shape Dağılımı için Optimal İlerleyen Tür Grup Sansürleme Planı”, Uluslararası 7. İstatistik Kongresi, Antalya,28 Nisan-01 Mayıs 2011., Sayı 4, 2011, No:1, 214-215 (2011).
• Kuş, C., Akdoğan, Y., Wu, S.-J., “Optimal progressive group-censoring plans for Burr XII distribution under cost constraint “, 5th Annual International Conference on Mathematics, Statistics & Mathematical Education, Athens, Greece, 13-16 June 2011, MAT2011/1012 (2011a).
• Kuş, C., Akdoğan, Y., Wu, S.-J., “Optimal progressive group-censoring plans for Pareto distribution under cost constraint”. Submitted paper (2011b).
• Lu, W, Tsai, T-R. “Interval censored sampling plans for the gamma lifetime model”. European Journal of Operational Research. 192, 116-124 (2009).
• Lui, K. J., Sterey, D., Pugh, J.K. “Sample size determination for grouped exponential observation: A cost function approach”. Biometrical Journal, 35, 677-688 (1993).
• Papadopoulos, A.S., “The Burr distribution as a failure model from a Bayesian approach”, IEEE Trans. Reliab. R-27, 367–371 (1978).
• Qian, L., Correa, J.A., “Estimation of Weibull parameters for grouped data with competing risks”, Journal of Staistical Computation and Simulation. 73, 261-275 (2003).
• Tse, S.-K.,Yang, C., Yuen, H. K., “Design and analysis of survival data under an integrated type-II interval Biopharmaceutical Statistics, 12, 333-345 (2002). Journal of
• Wu, S.-J., Chang, C.-T., Liao, K.-J., Huang, S.-R., “Planning of progressive group censoring life tests with cost considerations”, Journal of Applied Statistics, 35, 1293-1304 (2008a).
• Wu, S.-J., Lin, Y.-P., Chen, S.-T. “Optimal step-stress group-censoring with random removals”, Journal of Statistical Planning and Inference, 138, 817-826 (2008b). type I progressive
• Xiang, L., Tse, S.-K., “Maximum likelihood Estimation in survival studies under progressive interval censoring with random removals”, Journal of Biopharmaceutical Statistics. 15, 981-991 (2005).
• Yang, C., Tse, S.-K., “Planning accelerated life tests under progressive type-I Interval censoring with random removals”, Communications in Statistics -Simulation and Computation, 34, 1001-1025 (2005).
• Zimmer, W. J., Keats, J. B., Wang, F. K., “The Burr XII distribution in reliability analysis”, Journal of Quality Technology, 30, 386–394 (1998).