Bayesian Inference and Prediction of Burr Type XII Distribution for Progressive First Failure Censored Sampling

ABSTRACT

This paper deals with Bayesian inference and prediction problems of the Burr type XII distribution based on progressive first failure censored data. We consider the Bayesian inference under a squared error loss function. We propose to apply Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples, and they have in turn, been used to compute the Bayes estimates with the help of importance sampling technique. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We further consider two sample Bayes prediction to predicting future order statistics and upper record values from Burr type XII distribution based on progressive first failure censored data. The predictive densities are obtained and used to determine prediction intervals for unobserved order statistics and upper record values. A real life data set is used to illustrate the results derived.

This paper deals with Bayesian inference and prediction problems of the Burr type XII distribution based on progressive first failure censored data. We consider the Bayesian inference under a squared error loss function. We propose to apply Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples, and they have in turn, been used to compute the Bayes estimates with the help of importance sampling technique. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We further consider two sample Bayes prediction to predicting future order statistics and upper record values from Burr type XII distribution based on progressive first failure censored data. The predictive densities are obtained and used to determine prediction intervals for unobserved order statistics and upper record values. A real life data set is used to illustrate the results derived.

KEYWORDS

Burr Type XII Distribution, Progressive First-Failure Censored Sample, Bayesian Estimations, Gibbs Sampling, Markov Chain Monte Carlo, Posterior Predictive Density

Burr Type XII Distribution, Progressive First-Failure Censored Sample, Bayesian Estimations, Gibbs Sampling, Markov Chain Monte Carlo, Posterior Predictive Density

Cite this paper

nullA. Soliman, A. Ellah, N. Abou-Elheggag and A. Modhesh, "Bayesian Inference and Prediction of Burr Type XII Distribution for Progressive First Failure Censored Sampling,"*Intelligent Information Management*, Vol. 3 No. 5, 2011, pp. 175-185. doi: 10.4236/iim.2011.35021.

nullA. Soliman, A. Ellah, N. Abou-Elheggag and A. Modhesh, "Bayesian Inference and Prediction of Burr Type XII Distribution for Progressive First Failure Censored Sampling,"

References

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[2] M. Z. Raqab, A. R. Asgharzadeh and R. Valiollahi, “Prediction for Pareto distribu-tion based on progressively Type-II Censored Samples,” Com-putational Statistics and Data Analysis,” Vol. 54, No. 7, 2010, pp. 1732-1743. doi:10.1016/j.csda.2010.02.005

[3] N. Balakrishnan and R. Aggarwala, “Progressive Censoring: Theory, Methods, and Applications,” Birkhauser, Boston, 2000.

[4] N. Balakrishnan, “Progressive Censoring Methodology: An Appraisal,” Test, Vol. 16, No. 2, 2007, pp. 211-296. doi:10.1007/s11749-007-0061-y

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

[6] C.-H. Jun, S. Balamurali and S.-H. Lee, “Variables Sampling Plans for Weibull Distributed Lifetimes under Sud-den Death Testing,” IEEE Transactions on Reliability, Vol. 55, No. 1, 2006, pp. 53-58. doi:10.1109/TR.2005.863802

[7] J.-W. WU, W.-L. Hung and C.-H. Tsai, “Estimation of the Parameters of the Gompertz Distribution under the First-Failure-Censored Sampling Plan,” Statistics, Vol. 37, No. 6, 2003, pp. 517-525. doi:10.1080/02331880310001598864

[8] J.-W. Wu and H.-Y. Yu, “Statistical Inference about the Shape Parameter of the Burr Type XII Distribution under the Failure-Censored Sam-pling Plan,” Applied Mathematics and computation, Vol. 163, No. 1, 2005, pp. 443-482. doi:10.1016/j.amc.2004.02.019

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

[10] G. J. Hahn and W. Q. Meeker, “Statistical Intervals: A Guide for Practitioners,” John Wiley and Sons, Hoboken, 1991.

[11] A. M. Nigm, “Predic-tion Bounds for the Burr Model,” Communications in Statis-tics-Theory and Methods, Vol. 17, No. 1, 1988, pp. 287-297. doi:10.1080/03610928808829622

[12] E. K. AL-Huesaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr Type Xll Failure Model,” Communications in Statistics-Theory and Methods, Vol. 24, No. 7, 1995, pp. 1829-1842. doi:10.1080/03610929508831589

[13] E. K. AL-Huesaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr Type XII Distribution in the Presence of Outliers,” Journal of Statis-tical Planning and Inference, Vol. 55, 2996, pp. 23-37.

[14] M. A. M. Ali Mousa and Z. F. Jaheen, “Bayesian Prediction for the Burr Type XII Model Based on Doubly Censored Data,” Statis-tics, Vol. 48, 1997, pp. 337-344.

[15] M. A. M. Ali Mousa and Z. F. Jaheen, “ Bayesian Prediction for the Two-Paxameter Burr Type XII Model Based on Doubly Censored Data,” Jour-nal of Applied Statistical Science, Vol. 7, No. 2-3, 1998, pp. 103-111.

[16] Z. F. Jaheen and B. N. AL-Matrafi, “Bayesian Prediction Bounds from the Scaled Burr Type X Model,” Com-puters and Mathematics with Applications, Vol. 44, No. 5-6, 2002, pp. 587-594. doi:10.1016/S0898-1221(02)00173-6

[17] A. A. Alamm, M. Z. Raqab and M. T. Madi, “Bayesian Prediction Intervals for Fu-ture Order Statistics from the Generalized Exponential Distri-bution,” Journal of the Iranian Statistical Society, Vol. 6, No. 1, 2007, pp. 17-30.

[18] D. Kundu and H. Howlader, “Bayesian Inference and Prediction of the Inverse Weibull Distribution for Type-II Censored Data,” Computational Statistics and Data Analysis, Vol. 54, No. 6, 2010, pp. 1547 1558. doi:10.1016/j.csda.2010.01.003

[19] J. Ahmadi, S. M. T. K. MirMostafaee and N. Balakrishnanb, “Bayesian Prediction of Order Statistics Based on k-Record Values from Exponential Distribution,” Statistics, Vol. 44, No. 5, 2010, pp. 1-13.

[20] J. Ahmadi and S. M. T. K. MirMostafaee, “Prediction Intervals for Future Records and Order Statistics Coming from Two parameter Exponential Distribution,” Statistics and Probability Letters, Vol. 79, No. 7, 2009, pp. 977-983. doi:10.1016/j.spl.2008.12.002

[21] M. A. M. Ali Mousa and S. A. AL-Sagheer, “Bayesian Prediction for Progressively Type-II Censored Data from the Rayleigh Model,” Communications in Statistics-Theory and Methods, Vol. 34, No. 12, 2005, pp. 2353-2361. doi:10.1080/03610920500313767

[22] I. W. Burr, “Cumulative Frequency Functions,” Annals of Mathematical Statistics, Vol. 13, No. 2, 1942, pp. 215-232. doi:10.1214/aoms/1177731607

[23] R. N. Rodriguez, “A Guide to the Burr Type XII Distributions,” Biometrika, Vol. 64, No. 1, 1977, pp. 129-134. doi:10.1093/biomet/64.1.129

[24] W. C. Lee, J. W. Wu and C. W. Hong, “Assessing the Lifetime Performance Index of Prod-ucts from Progressively Type II Right Censored Data Using Burr XII Model,” Mathematics and Computers in Simulation, Vol. 79, No. 7, 2009, pp. 2167-2179. doi:10.1016/j.matcom.2008.12.001

[25] E. K. Al-Hussaini, “Predicting Observables from a General Class of Distribu-tions,” Journal of Statistical Planning and Inference, Vol. 79, No. 1, 1999, pp. 79-91. doi:10.1016/S0378-3758(98)00228-6

[26] M.-H. Chen and Q.-M. Shao, “Monte Carlo Estimation of Bayesian Credible and HPD Intervals,” Journal of Computational and Graphical Statistics, Vol. 8, No. 1, 1999, pp. 69-92. doi:10.2307/1390921

[27] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equations of State Calculations by Fast Computing Machines,” Journal Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1091. doi:10.1063/1.1699114

[28] D. R. Wingo, “Maximum Likeli-hood Methods for Fitting the Burr Type XII Distribution to Life Test Data,” Metrika, Vol. 40, No. 1, 1993, pp. 203-210.

[29] S.-F. Wu, C.-C. Wu, Y.-L. Chen, Y.-R. Yu and Y. P. Lin, “Interval Estimation of a Two-Parameter Burr-XII Distribution under progressive Censoring,” Statistics, Vol. 44, No. 1, 2010, pp. 77-88. doi:10.1080/02331880902757922

[1] D. Kundu, “Bayesian Inference and Life Testing Plan for the Weibull Distribution in Presence of Progressive Censoring,” Technometrics, Vol. 50, No. 2, 2008, pp. 144- 154. doi:10.1198/004017008000000217

[2] M. Z. Raqab, A. R. Asgharzadeh and R. Valiollahi, “Prediction for Pareto distribu-tion based on progressively Type-II Censored Samples,” Com-putational Statistics and Data Analysis,” Vol. 54, No. 7, 2010, pp. 1732-1743. doi:10.1016/j.csda.2010.02.005

[3] N. Balakrishnan and R. Aggarwala, “Progressive Censoring: Theory, Methods, and Applications,” Birkhauser, Boston, 2000.

[4] N. Balakrishnan, “Progressive Censoring Methodology: An Appraisal,” Test, Vol. 16, No. 2, 2007, pp. 211-296. doi:10.1007/s11749-007-0061-y

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

[6] C.-H. Jun, S. Balamurali and S.-H. Lee, “Variables Sampling Plans for Weibull Distributed Lifetimes under Sud-den Death Testing,” IEEE Transactions on Reliability, Vol. 55, No. 1, 2006, pp. 53-58. doi:10.1109/TR.2005.863802

[7] J.-W. WU, W.-L. Hung and C.-H. Tsai, “Estimation of the Parameters of the Gompertz Distribution under the First-Failure-Censored Sampling Plan,” Statistics, Vol. 37, No. 6, 2003, pp. 517-525. doi:10.1080/02331880310001598864

[8] J.-W. Wu and H.-Y. Yu, “Statistical Inference about the Shape Parameter of the Burr Type XII Distribution under the Failure-Censored Sam-pling Plan,” Applied Mathematics and computation, Vol. 163, No. 1, 2005, pp. 443-482. doi:10.1016/j.amc.2004.02.019

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

[10] G. J. Hahn and W. Q. Meeker, “Statistical Intervals: A Guide for Practitioners,” John Wiley and Sons, Hoboken, 1991.

[11] A. M. Nigm, “Predic-tion Bounds for the Burr Model,” Communications in Statis-tics-Theory and Methods, Vol. 17, No. 1, 1988, pp. 287-297. doi:10.1080/03610928808829622

[12] E. K. AL-Huesaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr Type Xll Failure Model,” Communications in Statistics-Theory and Methods, Vol. 24, No. 7, 1995, pp. 1829-1842. doi:10.1080/03610929508831589

[13] E. K. AL-Huesaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr Type XII Distribution in the Presence of Outliers,” Journal of Statis-tical Planning and Inference, Vol. 55, 2996, pp. 23-37.

[14] M. A. M. Ali Mousa and Z. F. Jaheen, “Bayesian Prediction for the Burr Type XII Model Based on Doubly Censored Data,” Statis-tics, Vol. 48, 1997, pp. 337-344.

[15] M. A. M. Ali Mousa and Z. F. Jaheen, “ Bayesian Prediction for the Two-Paxameter Burr Type XII Model Based on Doubly Censored Data,” Jour-nal of Applied Statistical Science, Vol. 7, No. 2-3, 1998, pp. 103-111.

[16] Z. F. Jaheen and B. N. AL-Matrafi, “Bayesian Prediction Bounds from the Scaled Burr Type X Model,” Com-puters and Mathematics with Applications, Vol. 44, No. 5-6, 2002, pp. 587-594. doi:10.1016/S0898-1221(02)00173-6

[17] A. A. Alamm, M. Z. Raqab and M. T. Madi, “Bayesian Prediction Intervals for Fu-ture Order Statistics from the Generalized Exponential Distri-bution,” Journal of the Iranian Statistical Society, Vol. 6, No. 1, 2007, pp. 17-30.

[18] D. Kundu and H. Howlader, “Bayesian Inference and Prediction of the Inverse Weibull Distribution for Type-II Censored Data,” Computational Statistics and Data Analysis, Vol. 54, No. 6, 2010, pp. 1547 1558. doi:10.1016/j.csda.2010.01.003

[19] J. Ahmadi, S. M. T. K. MirMostafaee and N. Balakrishnanb, “Bayesian Prediction of Order Statistics Based on k-Record Values from Exponential Distribution,” Statistics, Vol. 44, No. 5, 2010, pp. 1-13.

[20] J. Ahmadi and S. M. T. K. MirMostafaee, “Prediction Intervals for Future Records and Order Statistics Coming from Two parameter Exponential Distribution,” Statistics and Probability Letters, Vol. 79, No. 7, 2009, pp. 977-983. doi:10.1016/j.spl.2008.12.002

[21] M. A. M. Ali Mousa and S. A. AL-Sagheer, “Bayesian Prediction for Progressively Type-II Censored Data from the Rayleigh Model,” Communications in Statistics-Theory and Methods, Vol. 34, No. 12, 2005, pp. 2353-2361. doi:10.1080/03610920500313767

[22] I. W. Burr, “Cumulative Frequency Functions,” Annals of Mathematical Statistics, Vol. 13, No. 2, 1942, pp. 215-232. doi:10.1214/aoms/1177731607

[23] R. N. Rodriguez, “A Guide to the Burr Type XII Distributions,” Biometrika, Vol. 64, No. 1, 1977, pp. 129-134. doi:10.1093/biomet/64.1.129

[24] W. C. Lee, J. W. Wu and C. W. Hong, “Assessing the Lifetime Performance Index of Prod-ucts from Progressively Type II Right Censored Data Using Burr XII Model,” Mathematics and Computers in Simulation, Vol. 79, No. 7, 2009, pp. 2167-2179. doi:10.1016/j.matcom.2008.12.001

[25] E. K. Al-Hussaini, “Predicting Observables from a General Class of Distribu-tions,” Journal of Statistical Planning and Inference, Vol. 79, No. 1, 1999, pp. 79-91. doi:10.1016/S0378-3758(98)00228-6

[26] M.-H. Chen and Q.-M. Shao, “Monte Carlo Estimation of Bayesian Credible and HPD Intervals,” Journal of Computational and Graphical Statistics, Vol. 8, No. 1, 1999, pp. 69-92. doi:10.2307/1390921

[27] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equations of State Calculations by Fast Computing Machines,” Journal Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1091. doi:10.1063/1.1699114

[28] D. R. Wingo, “Maximum Likeli-hood Methods for Fitting the Burr Type XII Distribution to Life Test Data,” Metrika, Vol. 40, No. 1, 1993, pp. 203-210.

[29] S.-F. Wu, C.-C. Wu, Y.-L. Chen, Y.-R. Yu and Y. P. Lin, “Interval Estimation of a Two-Parameter Burr-XII Distribution under progressive Censoring,” Statistics, Vol. 44, No. 1, 2010, pp. 77-88. doi:10.1080/02331880902757922