Bayesian and Non-Bayesian Estimation of the Inverse Weibull Model Based on Generalized Order Statistics

Author(s)
Ahmed H. Abd Ellah

ABSTRACT

The concept of generalized order statistics has been introduced as a unified approach to a variety of models of ordered random variables with different interpretations. In this paper, we develop methodology for constructing inference based on n selected generalized order statistics (GOS) from inverse Weibull distribution (IWD), Bayesian and non-Bayesian approaches have been used to obtain the estimators of the parameters and reliability function. We have examined Bayes estimates under various losses such as the balanced squared error (balanced SEL) and balanced LINEX loss functions are considered. We show that Bayes estimate under balanced SEL and balanced LINEX loss functions are more general, which include the symmetric and asymmetric losses as special cases. This was done under assumption of discrete-continuous mixture prior for the unknown model parameters. The parametric bootstrap method has been used to construct confidence interval for the parameters and reliability function. Progressively type-II censored and k-record values as a special case of GOS are considered. Finally a practical example using real data set was used for illustration.

The concept of generalized order statistics has been introduced as a unified approach to a variety of models of ordered random variables with different interpretations. In this paper, we develop methodology for constructing inference based on n selected generalized order statistics (GOS) from inverse Weibull distribution (IWD), Bayesian and non-Bayesian approaches have been used to obtain the estimators of the parameters and reliability function. We have examined Bayes estimates under various losses such as the balanced squared error (balanced SEL) and balanced LINEX loss functions are considered. We show that Bayes estimate under balanced SEL and balanced LINEX loss functions are more general, which include the symmetric and asymmetric losses as special cases. This was done under assumption of discrete-continuous mixture prior for the unknown model parameters. The parametric bootstrap method has been used to construct confidence interval for the parameters and reliability function. Progressively type-II censored and k-record values as a special case of GOS are considered. Finally a practical example using real data set was used for illustration.

Cite this paper

A. Abd Ellah, "Bayesian and Non-Bayesian Estimation of the Inverse Weibull Model Based on Generalized Order Statistics,"*Intelligent Information Management*, Vol. 4 No. 2, 2012, pp. 23-31. doi: 10.4236/iim.2012.42004.

A. Abd Ellah, "Bayesian and Non-Bayesian Estimation of the Inverse Weibull Model Based on Generalized Order Statistics,"

References

[1] U. Kamps, “A concept of Generalized Order Statistics,” Teubner Stuttgart, 1995,

[2] U. Kamps and E. Cramer, “On Dis-tributions of Generalized Order Statistics,” Statistics, Vol. 35, No. 3, 2001, pp. 269-280. doi:10.1080/02331880108802736

[3] H. A. David, “Order Statistics,” 2nd Edition, Wiley, New York, 1981.

[4] E. Castillo, “Extreme Value Theory in Engineering,” Academic Press, Boston, 1988.

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

[6] E. Cramer and U. Kamps, “Sequential Order Statistics and k-Out-of-n Systems with Sequentially Adjusted Failure Rates,” Annals of Institute of Statistical Mathematics, Vol. 48, No. 3, 1996, pp. 535-549. doi:10.1007/BF00050853

[7] E. Cramer and U. Kamps, “Marginal Distributions of Sequential and Generalized Order Statistics,” Metrika, Vol. 58, No. 2, 2003, pp. 293-310. doi:10.1007/s001840300268

[8] V. B. Nevzorov, “Records,” Theory of Probability and Its Applications, Vol. 32, No. 2, 1987, pp. 201-228. doi:10.1137/1132032

[9] M. Ahsanullah, “Record Statistics,” Nova Science Publisher, Inc., Commack, New York, 1995.

[10] A. A. Soliman and G. R. Elkahlout, “Bayes Estimation of the Logistic Distribution Based on Progressively Cen-sored Samples,” Journal of Applied Statistical Science, Vol. 14, 2005, pp. 281-293.

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

[12] A. A. Soliman, “Estimations for Pareto Model Using General Progressive Censored Data and Asymmetric Loss,” Communications in Statistics Theory & Methods, Vol. 37, No. 9, 2008, pp. 1353-1370. doi:10.1080/03610920701825957

[13] A. A., Soliman, A. H. Abd Ellah, N. A. Abou-Elheggag and A. A. Modhesh, “Bayesian Inference and Prediction of Burr Type XII Distribution for Progressive First Failure Censored Sampling,” Intelligent Information Management, Vol. 3, 2011, pp. 175-185.

[14] N. Balakrishnan and A. Asgharzadeh, “Inference for the Scaled Half-Logistic Distribution Based on Progressively Type-II Censored Samples,” Communications in Statistics Theory & Methods, Vol. 34 , No. 1, 2005, pp. 73-87.

[15] A. M. Sarhan and A. Abuammoh, “Statistical Inference Using Progressively Type-II Censored Data with Random Scheme,” Int. Math. Forum, Vol. 3, No. 33-36, 2008, pp. 1713-1725.

[16] A. S. Wahed, “Bayesian Inference Using Burr Model Under Asymmetric Loss Function: An Application to Carcinoma Survival Data,” Journal of Statistical Research, Vol. 40, No. 1, 2006, pp. 45-57.

[17] J. R. Alicja, “A Sequential Estimation Procedure for the Parameter of an Exponential Distribution under Asym-metric Loss Function,” Statistics & Probability Letters, Vol. 78, No. 17, 2008, pp. 3091-3095.

[18] A. H. Abd Ellah, “Bayesian Prediction of Weibull Distributions Based on Fixed and Random Sample size,” Serdica Mathematical Journal, Vol. 35, 2009, pp. 129-146.

[19] A. H. Abd Ellah, “Parametric Predic-tion Limit for Generalized Exponential Distribution Using Re-cord Observations,” Applied Mathematics & Information Sci-ence, No. 2, 2008, pp. 135-149.

[20] A. H. Abd Ellah, “Com-parison of Estimates Using Record Statistics from Lomax Model: Bayesian and Non- Bayesian Approaches,” Journal of Statistical Research of Iran Statistical Research and Training Center, Vol. 3, No. 2, 2006, pp. 139-158.

[21] K. S. Sultan, “Bayesian Estimates Based on Record Values from the Inverse Weibull Lifetime,” Model Quality Technology & Quantitative Management, Vol. 5, No. 4, 2008, pp. 363-374.

[22] W. B. Nelson, “Applied Life Data Analysis,” John Wiley & Sons, New York, 1982.

[23] R. Calabria and G. Pulcini, “On the Maximum Likelihood and Least Squares Estimation in the In-verse Weibull Distribution,” Statistica Applicata, Vol. 2, No. 1, 1990, pp. 53-66.

[24] M. Maswadah, “Conditional Confidence Interval Estimation for the Inverse Weibull Distribution Based on Censored Generalized Order Statistics,” Journal of Statistical Computation and Simulation, Vol. 73, No. 12, 2003, pp. 887-898. doi:10.1080/0094965031000099140

[25] R. Dumonceaux and C. E. Antle, “Discrimination between the Lognormal and Weibull Distribution,” Technometrics, Vol. 15, 1973, pp. 923-926. doi:10.2307/1267401

[26] D. N. P. Murthy, M. Xie and R. Jiang, “Weibull Model,” John Wiley & Sons, New York, 2004.

[27] J. Ahmadi, M. J. Jozani, E. Marchand and A. Par-sian, “Bayes Estimation Based on k-Record Data from a Gen-eral Class of Distributions under Balanced Type Loss Func-tions,” Journal of Statistical Planning and Inference, Vol. 139, No. 3, 2009, pp. 1180-1189.

[28] J. F. Lawless, “Statistical Models and Methods for Lifetime Data,” John Wiley & Sons, New York, 2003.

[29] R. Aggarwala and N. Balakrishnan, “Maximum Likelihood Estimation of Laplace Parameters Based on Progressive Type-II Censored Samples,” In: N. Balakrishnan, Ed., Advances in Methods and Applications of Probability and Statistics, Gordon & Breach Publishers, New York, 1999.

[30] H. K. T. Ng, P. S. Chan and N. Balakrishnan, “Optimal Progressive Censoring Plans for the Weibull Distri-bution,” Technometics, Vol. 46, No. 4, 2004, pp. 470-481. doi:10.1198/004017004000000482

[31] N. Balakrishnan, N. Kannan, C. T. Lin and H. K. T. Ng, “Point and Interval Estima-tion for Gaussian Distribution Based on Progressively Type-II Censored Samples,” IEEE Transactions on Reliability, Vol. 52, No. 1, 2003, pp. 90-95. doi:10.1109/TR.2002.805786

[32] N. Balakrishnan, “Progressive Censoring Methodology: An Ap-praisal,” Test, Vol. 16, No. 2, 2007, pp. 211-259. doi:10.1007/s11749-007-0061-y

[33] N. Balakrishnan and R. A. Sandhu, “A Simple Simulational Algorithm for Generating Progressive Type-II Censored Samples,” American Statistician, Vol. 49, No. 2, 1995, pp. 229-230. doi:10.2307/2684646

[34] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, “Record,” NewYork, Wiley, 1998.

[35] B. C. Arnold and S. J. Press, “Bayesian Estimation and Prediction for Pareto Data,” Journal of the Acoustical Society of America (JASA), Vol. 84, 1998, pp. 1079-1084.

[36] M. Burkschat, E. Cramer and U. Kamps, “Linear Estimation of Location and Scale Parameters Based on Generalized Order Statistics from Generalized Pareto Distributions,” In: M. Ahsanullah, Eds., Recent Developments in Ordered Random Variables, Nova Science Publisher, New York, 2007, pp. 253-261.

[37] R. M. Soland, “Bayesian Analysis of the Weibull Process with Un-known Scale and Shape Parameters,” IEEE Transactions on Reliability, Vol. R18, No. 4, 1969, pp. 181- 184. doi:10.1109/TR.1969.5216348

[38] B. Efron, “Censored Data and Bootstrap,” Journal of the American Statistical Association, Vol. 76, No. 374, 1981, pp. 312-319. doi:10.2307/2287832

[39] B. Efron and R. J. Tibshirani, “Bootstrap Method for Standard Errors, Confidence Intervals and Other Measures of Statistical Accuracy,” Statistical Science, Vol. 1, No. 1, 1986, pp. 54-75. doi:10.1214/ss/1177013815

[40] H. F. Martz and R. A. Waller, “Bayesian Reliability Analysis,” John Wiley & Sons, New York, 1982.

[1] U. Kamps, “A concept of Generalized Order Statistics,” Teubner Stuttgart, 1995,

[2] U. Kamps and E. Cramer, “On Dis-tributions of Generalized Order Statistics,” Statistics, Vol. 35, No. 3, 2001, pp. 269-280. doi:10.1080/02331880108802736

[3] H. A. David, “Order Statistics,” 2nd Edition, Wiley, New York, 1981.

[4] E. Castillo, “Extreme Value Theory in Engineering,” Academic Press, Boston, 1988.

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

[6] E. Cramer and U. Kamps, “Sequential Order Statistics and k-Out-of-n Systems with Sequentially Adjusted Failure Rates,” Annals of Institute of Statistical Mathematics, Vol. 48, No. 3, 1996, pp. 535-549. doi:10.1007/BF00050853

[7] E. Cramer and U. Kamps, “Marginal Distributions of Sequential and Generalized Order Statistics,” Metrika, Vol. 58, No. 2, 2003, pp. 293-310. doi:10.1007/s001840300268

[8] V. B. Nevzorov, “Records,” Theory of Probability and Its Applications, Vol. 32, No. 2, 1987, pp. 201-228. doi:10.1137/1132032

[9] M. Ahsanullah, “Record Statistics,” Nova Science Publisher, Inc., Commack, New York, 1995.

[10] A. A. Soliman and G. R. Elkahlout, “Bayes Estimation of the Logistic Distribution Based on Progressively Cen-sored Samples,” Journal of Applied Statistical Science, Vol. 14, 2005, pp. 281-293.

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

[12] A. A. Soliman, “Estimations for Pareto Model Using General Progressive Censored Data and Asymmetric Loss,” Communications in Statistics Theory & Methods, Vol. 37, No. 9, 2008, pp. 1353-1370. doi:10.1080/03610920701825957

[13] A. A., Soliman, A. H. Abd Ellah, N. A. Abou-Elheggag and A. A. Modhesh, “Bayesian Inference and Prediction of Burr Type XII Distribution for Progressive First Failure Censored Sampling,” Intelligent Information Management, Vol. 3, 2011, pp. 175-185.

[14] N. Balakrishnan and A. Asgharzadeh, “Inference for the Scaled Half-Logistic Distribution Based on Progressively Type-II Censored Samples,” Communications in Statistics Theory & Methods, Vol. 34 , No. 1, 2005, pp. 73-87.

[15] A. M. Sarhan and A. Abuammoh, “Statistical Inference Using Progressively Type-II Censored Data with Random Scheme,” Int. Math. Forum, Vol. 3, No. 33-36, 2008, pp. 1713-1725.

[16] A. S. Wahed, “Bayesian Inference Using Burr Model Under Asymmetric Loss Function: An Application to Carcinoma Survival Data,” Journal of Statistical Research, Vol. 40, No. 1, 2006, pp. 45-57.

[17] J. R. Alicja, “A Sequential Estimation Procedure for the Parameter of an Exponential Distribution under Asym-metric Loss Function,” Statistics & Probability Letters, Vol. 78, No. 17, 2008, pp. 3091-3095.

[18] A. H. Abd Ellah, “Bayesian Prediction of Weibull Distributions Based on Fixed and Random Sample size,” Serdica Mathematical Journal, Vol. 35, 2009, pp. 129-146.

[19] A. H. Abd Ellah, “Parametric Predic-tion Limit for Generalized Exponential Distribution Using Re-cord Observations,” Applied Mathematics & Information Sci-ence, No. 2, 2008, pp. 135-149.

[20] A. H. Abd Ellah, “Com-parison of Estimates Using Record Statistics from Lomax Model: Bayesian and Non- Bayesian Approaches,” Journal of Statistical Research of Iran Statistical Research and Training Center, Vol. 3, No. 2, 2006, pp. 139-158.

[21] K. S. Sultan, “Bayesian Estimates Based on Record Values from the Inverse Weibull Lifetime,” Model Quality Technology & Quantitative Management, Vol. 5, No. 4, 2008, pp. 363-374.

[22] W. B. Nelson, “Applied Life Data Analysis,” John Wiley & Sons, New York, 1982.

[23] R. Calabria and G. Pulcini, “On the Maximum Likelihood and Least Squares Estimation in the In-verse Weibull Distribution,” Statistica Applicata, Vol. 2, No. 1, 1990, pp. 53-66.

[24] M. Maswadah, “Conditional Confidence Interval Estimation for the Inverse Weibull Distribution Based on Censored Generalized Order Statistics,” Journal of Statistical Computation and Simulation, Vol. 73, No. 12, 2003, pp. 887-898. doi:10.1080/0094965031000099140

[25] R. Dumonceaux and C. E. Antle, “Discrimination between the Lognormal and Weibull Distribution,” Technometrics, Vol. 15, 1973, pp. 923-926. doi:10.2307/1267401

[26] D. N. P. Murthy, M. Xie and R. Jiang, “Weibull Model,” John Wiley & Sons, New York, 2004.

[27] J. Ahmadi, M. J. Jozani, E. Marchand and A. Par-sian, “Bayes Estimation Based on k-Record Data from a Gen-eral Class of Distributions under Balanced Type Loss Func-tions,” Journal of Statistical Planning and Inference, Vol. 139, No. 3, 2009, pp. 1180-1189.

[28] J. F. Lawless, “Statistical Models and Methods for Lifetime Data,” John Wiley & Sons, New York, 2003.

[29] R. Aggarwala and N. Balakrishnan, “Maximum Likelihood Estimation of Laplace Parameters Based on Progressive Type-II Censored Samples,” In: N. Balakrishnan, Ed., Advances in Methods and Applications of Probability and Statistics, Gordon & Breach Publishers, New York, 1999.

[30] H. K. T. Ng, P. S. Chan and N. Balakrishnan, “Optimal Progressive Censoring Plans for the Weibull Distri-bution,” Technometics, Vol. 46, No. 4, 2004, pp. 470-481. doi:10.1198/004017004000000482

[31] N. Balakrishnan, N. Kannan, C. T. Lin and H. K. T. Ng, “Point and Interval Estima-tion for Gaussian Distribution Based on Progressively Type-II Censored Samples,” IEEE Transactions on Reliability, Vol. 52, No. 1, 2003, pp. 90-95. doi:10.1109/TR.2002.805786

[32] N. Balakrishnan, “Progressive Censoring Methodology: An Ap-praisal,” Test, Vol. 16, No. 2, 2007, pp. 211-259. doi:10.1007/s11749-007-0061-y

[33] N. Balakrishnan and R. A. Sandhu, “A Simple Simulational Algorithm for Generating Progressive Type-II Censored Samples,” American Statistician, Vol. 49, No. 2, 1995, pp. 229-230. doi:10.2307/2684646

[34] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, “Record,” NewYork, Wiley, 1998.

[35] B. C. Arnold and S. J. Press, “Bayesian Estimation and Prediction for Pareto Data,” Journal of the Acoustical Society of America (JASA), Vol. 84, 1998, pp. 1079-1084.

[36] M. Burkschat, E. Cramer and U. Kamps, “Linear Estimation of Location and Scale Parameters Based on Generalized Order Statistics from Generalized Pareto Distributions,” In: M. Ahsanullah, Eds., Recent Developments in Ordered Random Variables, Nova Science Publisher, New York, 2007, pp. 253-261.

[37] R. M. Soland, “Bayesian Analysis of the Weibull Process with Un-known Scale and Shape Parameters,” IEEE Transactions on Reliability, Vol. R18, No. 4, 1969, pp. 181- 184. doi:10.1109/TR.1969.5216348

[38] B. Efron, “Censored Data and Bootstrap,” Journal of the American Statistical Association, Vol. 76, No. 374, 1981, pp. 312-319. doi:10.2307/2287832

[39] B. Efron and R. J. Tibshirani, “Bootstrap Method for Standard Errors, Confidence Intervals and Other Measures of Statistical Accuracy,” Statistical Science, Vol. 1, No. 1, 1986, pp. 54-75. doi:10.1214/ss/1177013815

[40] H. F. Martz and R. A. Waller, “Bayesian Reliability Analysis,” John Wiley & Sons, New York, 1982.