Bayesian and Frequentist Prediction Using Progressive Type-II Censored with Binomial Removals

Affiliation(s)

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

Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt.

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

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

Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt.

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

ABSTRACT

In this article, we study the problem of predicting future records and order statistics (two-sample prediction) based on progressive type-II censored with random removals, where the number of units removed at each failure time has a discrete binomial distribution. We use the Bayes procedure to derive both point and interval bounds prediction. Bayesian point prediction under symmetric and symmetric loss functions is discussed. The maximum likelihood (ML) prediction intervals using “plug-in” procedure for future records and order statistics are derived. An example is discussed to illustrate the application of the results under this censoring scheme.

In this article, we study the problem of predicting future records and order statistics (two-sample prediction) based on progressive type-II censored with random removals, where the number of units removed at each failure time has a discrete binomial distribution. We use the Bayes procedure to derive both point and interval bounds prediction. Bayesian point prediction under symmetric and symmetric loss functions is discussed. The maximum likelihood (ML) prediction intervals using “plug-in” procedure for future records and order statistics are derived. An example is discussed to illustrate the application of the results under this censoring scheme.

Cite this paper

A. Soliman, A. Abd Ellah, N. Abou-Elheggag and R. El-Sagheer, "Bayesian and Frequentist Prediction Using Progressive Type-II Censored with Binomial Removals,"*Intelligent Information Management*, Vol. 5 No. 5, 2013, pp. 162-170. doi: 10.4236/iim.2013.55017.

A. Soliman, A. Abd Ellah, N. Abou-Elheggag and R. El-Sagheer, "Bayesian and Frequentist Prediction Using Progressive Type-II Censored with Binomial Removals,"

References

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[5] E. K. Al-Hussaini and Z. F. Jaheen, “Parametric Prediction Bounds for the Future Median of the Exponential Distribution,” Statistics, Vol. 32, No. 3, 1999, pp. 267-275. doi:10.1080/02331889908802667

[6] I. G. Evans and A. M. Nigm, “Bayesian Prediction for the Left Truncated Exponential Distribution,” Technometrics, Vol. 22, No. 2, 1980, pp. 201-204. doi:10.1080/00401706.1980.10486135

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[10] E. K. Al-Hussaini and Z. F. Jaheen, “Bayesian Prediction Bounds for Burr Type XII Distribution in the Presence of Outliers,” Statistical Planning and Inference, Vol. 55, No. 1, 1996, pp. 23-37. doi:10.1016/0378-3758(95)00184-0

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[13] J. K. Patel, “Prediction Intervals—A Review,” Communication in Statistics Theory & Methods, Vol. 18, No. 7, 1989, pp. 2393-2465.

[14] H. N. Nagaraja, “Prediction Problems,” In: N. Balakrishnan and A. P. Basu, Eds., The Exponential Distribution: Theory and Applications, Gordon and Breach, New York, 1995, pp. 139-163.

[15] K. S. Kaminsky and P. I. Nelson, “Prediction on Order Statistics,” In: N. Balakrishnan and C. R. Rao, Eds., Handbook of Statistics, Elsevier Science, Amsterdam, 1998, pp. 431-450.

[16] E. K. Al-Hussaini, “Prediction: Advances and New Research,” International Conference of Mathematics and 21st Century, Cairo, 15-22 January 2000, pp. 25-33.

[17] J. Aitchison and I. R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975. doi:10.1017/CBO9780511569647

[18] S. Geisser, “Predictive Inference: An Introduction,” Chapman and Hall, 1993.

[19] M. A. M. Ali Mousa, “Inference and Prediction for the Burr Type X Model Based on Records,” Statistics, Vol. 35, No. 4, 2001, pp. 415-425. doi:10.1080/02331880108802745

[20] Z. F. Jaheen and B. N. AL-Matrafi, “Bayesian Prediction Bounds from the Scaled Burr Type X Model,” Computers and Mathematics, Vol. 44, No. 5, 2002, pp. 587-594.

[21] A. A. Soliman, “Bayesian Prediction with Outliers and Random Sampel Size for the Burr Type X Model,” Mathematics & Physics Socity, Vol. 73, No. 1, 1998, pp. 1-12.

[22] H. A. Sartawi and M. S. Abu-Salih, “Bayesian Prediction Bounds for Burr Type X Model,” Communication in Statistics Theory & Methods, Vol. 20, No. 7, 1991, pp. 2307-2330.

[23] J. Ahmadi and N. Balakrishnan, “Prediction of Order Statistics and Record Values from Two Independent Sequences,” Journal of Theoretical and Applied Statistics, Vol. 44, No. 4, 2010, pp. 417-430.

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

[25] J. Ahmadi, S. M. T. K. Mir Mostafaee and N. Balakrishnanb, “Bayesian Prediction of Order Statistics Based on k-Record Values from Exponential Distribution,” Statistics, Vol. 45, No. 4, 2011, pp. 375-387.

[26] B .C. Arnold, N. Balakrishnan and H. N. Nagaraja, “A First Course in Order Statistics,” Wiley, New York, 1992.

[27] H. A. David and H. N. Nagaraja, “Order Statistics,” Wiley, New York, 2003.

[1] G. J. Hahan and W.Q. Meeker, “Statistical Interval: A Guide for Practitioners,” Johan Wiley and Sons, Hoboken, 1991. doi:10.1002/9780470316771

[2] I. R. Dunsmore, “The Bayesian Predictive Distribution in Life Testing Models,” Technometrics, Vol. 16, No. 3, 1974, pp. 455-460. doi:10.1080/00401706.1974.10489216

[3] G. S. Lingappaiah, “Bayesian Approach to Prediction and the Spacings in the Exponential Distribution,” Annals Institute of Statistics & Mathematics, Vol. 31, No. 1, 1979, pp. 391-401.

[4] I. G. Evans and A. M. Nigm, “Bayesian One-Sample Prediction for the Two-Parameter Weibull Distribution,” IEEE Transection, Vol. 29, No. 2, 1980, pp. 410-413.

[5] E. K. Al-Hussaini and Z. F. Jaheen, “Parametric Prediction Bounds for the Future Median of the Exponential Distribution,” Statistics, Vol. 32, No. 3, 1999, pp. 267-275. doi:10.1080/02331889908802667

[6] I. G. Evans and A. M. Nigm, “Bayesian Prediction for the Left Truncated Exponential Distribution,” Technometrics, Vol. 22, No. 2, 1980, pp. 201-204. doi:10.1080/00401706.1980.10486135

[7] I. G. Evans and A. M. Nigm, “Bayesian Prediction for Two-Parameter Weibull Lifetime Model,” Communication in Statistics Theory & Methods, Vol. 9, No. 6, 1980, pp. 649-658. doi:10.1080/03610928008827909

[8] A. M. Nigm, “Prediction Bounds for the Burr Model,” Communication in Statistics Theory & Methods, Vol. 17, No. 1, 1988, pp. 287-297. doi:10.1080/03610928808829622

[9] E. K. Al-Hussaini and Z. F. Jaheen, “Bayesian Prediction Bounds for Burr Type XI Model,” Communication in Statistics Theory & Methods, Vol. 24, No. 7, 1995, pp. 1829-1842.

[10] E. K. Al-Hussaini and Z. F. Jaheen, “Bayesian Prediction Bounds for Burr Type XII Distribution in the Presence of Outliers,” Statistical Planning and Inference, Vol. 55, No. 1, 1996, pp. 23-37. doi:10.1016/0378-3758(95)00184-0

[11] M. A. M. Ali Mousa and Z. F. Jaheen, “Bayesian Prediction Bounds for Burr Type XII Model Based on Doubly Censored Data,” Statistics, Vol. 48, No. 2, 1997, pp. 337-344.

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

[13] J. K. Patel, “Prediction Intervals—A Review,” Communication in Statistics Theory & Methods, Vol. 18, No. 7, 1989, pp. 2393-2465.

[14] H. N. Nagaraja, “Prediction Problems,” In: N. Balakrishnan and A. P. Basu, Eds., The Exponential Distribution: Theory and Applications, Gordon and Breach, New York, 1995, pp. 139-163.

[15] K. S. Kaminsky and P. I. Nelson, “Prediction on Order Statistics,” In: N. Balakrishnan and C. R. Rao, Eds., Handbook of Statistics, Elsevier Science, Amsterdam, 1998, pp. 431-450.

[16] E. K. Al-Hussaini, “Prediction: Advances and New Research,” International Conference of Mathematics and 21st Century, Cairo, 15-22 January 2000, pp. 25-33.

[17] J. Aitchison and I. R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975. doi:10.1017/CBO9780511569647

[18] S. Geisser, “Predictive Inference: An Introduction,” Chapman and Hall, 1993.

[19] M. A. M. Ali Mousa, “Inference and Prediction for the Burr Type X Model Based on Records,” Statistics, Vol. 35, No. 4, 2001, pp. 415-425. doi:10.1080/02331880108802745

[20] Z. F. Jaheen and B. N. AL-Matrafi, “Bayesian Prediction Bounds from the Scaled Burr Type X Model,” Computers and Mathematics, Vol. 44, No. 5, 2002, pp. 587-594.

[21] A. A. Soliman, “Bayesian Prediction with Outliers and Random Sampel Size for the Burr Type X Model,” Mathematics & Physics Socity, Vol. 73, No. 1, 1998, pp. 1-12.

[22] H. A. Sartawi and M. S. Abu-Salih, “Bayesian Prediction Bounds for Burr Type X Model,” Communication in Statistics Theory & Methods, Vol. 20, No. 7, 1991, pp. 2307-2330.

[23] J. Ahmadi and N. Balakrishnan, “Prediction of Order Statistics and Record Values from Two Independent Sequences,” Journal of Theoretical and Applied Statistics, Vol. 44, No. 4, 2010, pp. 417-430.

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

[25] J. Ahmadi, S. M. T. K. Mir Mostafaee and N. Balakrishnanb, “Bayesian Prediction of Order Statistics Based on k-Record Values from Exponential Distribution,” Statistics, Vol. 45, No. 4, 2011, pp. 375-387.

[26] B .C. Arnold, N. Balakrishnan and H. N. Nagaraja, “A First Course in Order Statistics,” Wiley, New York, 1992.

[27] H. A. David and H. N. Nagaraja, “Order Statistics,” Wiley, New York, 2003.