Estimation of the Unknown Parameters for the Compound Rayleigh Distribution Based on Progressive First-Failure-Censored Sampling

Author(s)
Tahani A. Abushal

ABSTRACT

This article considers estimation of the unknown parameters for the compound Rayleigh distribution (CRD) based on a new life test plan called a progressive first failure-censored plan introduced by Wu and Kus (2009). We consider the maximum likelihood and Bayesian inference of the unknown parameters of the model, as well as the reliability and hazard rate functions. This was done using the conjugate prior for the shape parameter, and discrete prior for the scale parameter. The Bayes estimators hav been obtained relative to both symmetric (squared error) and asymmetric (LINEX and general entropy (GE)) loss functions. It has been seen that the symmetric and asymmetric Bayes estimators are obtained in closed forms. Also, based on this new censoring scheme, approximate confidence intervals for the parameters of CRD are developed. A practical example using real data set was used for illustration. Finally, to assess the performance of the proposed estimators, some numerical results using Monte Carlo simulation study were reported.

This article considers estimation of the unknown parameters for the compound Rayleigh distribution (CRD) based on a new life test plan called a progressive first failure-censored plan introduced by Wu and Kus (2009). We consider the maximum likelihood and Bayesian inference of the unknown parameters of the model, as well as the reliability and hazard rate functions. This was done using the conjugate prior for the shape parameter, and discrete prior for the scale parameter. The Bayes estimators hav been obtained relative to both symmetric (squared error) and asymmetric (LINEX and general entropy (GE)) loss functions. It has been seen that the symmetric and asymmetric Bayes estimators are obtained in closed forms. Also, based on this new censoring scheme, approximate confidence intervals for the parameters of CRD are developed. A practical example using real data set was used for illustration. Finally, to assess the performance of the proposed estimators, some numerical results using Monte Carlo simulation study were reported.

KEYWORDS

Compound Rayleigh Distribution, Progressive First-Failure Censored Scheme, Bayesian and Non-Bayesian Estimations, Approximate Confidence Intervals

Compound Rayleigh Distribution, Progressive First-Failure Censored Scheme, Bayesian and Non-Bayesian Estimations, Approximate Confidence Intervals

Cite this paper

nullT. Abushal, "Estimation of the Unknown Parameters for the Compound Rayleigh Distribution Based on Progressive First-Failure-Censored Sampling,"*Open Journal of Statistics*, Vol. 1 No. 3, 2011, pp. 161-171. doi: 10.4236/ojs.2011.13020.

nullT. Abushal, "Estimation of the Unknown Parameters for the Compound Rayleigh Distribution Based on Progressive First-Failure-Censored Sampling,"

References

[1] R. Viveros, N. Balakrishnan, “Interval Estimation of Pa- rameters of Life from Progressively Censored Data,” Te- chnometrics, Vol. 36, No. 1, 1994, pp. 84-91. doi:10.2307/1269201

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

[3] N. Balakrishnan and R. A. Sandhu, “Best Linear Unbiased and Maximum Likelihood Estimation for Exponential Distributions under General Progressive Type-II Censored Samples,” Sankhya: The Indian Journal of Seriestics, Vol. 58, No. 1, 1996, pp. 1-9.

[4] P. Mostert, J. Roux and A. Bekker, “Bayes Estimators of the Life Time Parameters Using the Compound Rayleigh model,” Journal of South African Statistical Association, Vol. 33, No. 2 ,1999, pp. 117-138.

[5] 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

[6] A. Bekker, J. Roux and P. Mostert, “A Generalization of the Compound Rayleigh Distribution: Using a Bayesian Methods on Cancer Survival Times,” Communications in Statistics—Theory and Methods, Vol. 29, No. 7, 2000, pp. 1419-1433. doi:10.1080/03610920008832554

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

[8] H. K. T. Ng, “Parameter estimation for a modifiied Wei- bull Distribution, for Progressively Type-II Censored Samples,” IEEE Transactions Reliability, Vol. 54, No. 3, 2005, pp. 374-380. doi:10.1109/TR.2005.853036

[9] N. Balakrishnan, N. Kannan, C. T. Lin and H. K. T. Ng, “Point and Interval Estimation 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

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

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

[12] D. G. Chen and Y. L. Lio, “Parameter Estimations for Generalized Exponential Distribution under Progressive Type-I Interval Censoring,” Computational Statistics and Data Analysis, Vol. 54, No. 6, 2010, pp. 1581-1591. doi:10.1016/j.csda.2010.01.007

[13] N. Balakrishnan and R. Aggarwala, “Progressive Censoring—Theory, Methods and Applications,” Birkh?user, Boston, 2000.

[14] N. Balakrishnan, “Progressive Censoring Methodology: An Appraisal,” Test, Vol. 16, No. 2007, pp. 211-289. doi:10.1007/s11749-007-0061-y

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

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

[17] 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

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

[19] J.-W. Wu, T.-R. Tsai and L.-Y. Ouyang, “Limited Failure-Censored Life Test for the Weibull Distribution,” IEEE Transactions on Reliability, Vol. 50, No. 1, 2001, pp. 107-111. doi:10.1109/24.935024

[20] W.-C. Lee, L.-W. Wu and H.-Y. Yu, “Statistical Inference about the Shape Parameter of the Bathtub-Shaped Distribution under the Failure-Censored Sampling Plan,” International Journal of Information and Management Sciences, Vol. 18, No. 2, 2007, pp. 157-172.

[21] 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. 1-12.

[22] E. K. AL-Hussaini and Z. F. Jaheen, “Bayes Estimation of the Parameters, Reliability and Failure Rate Functions of the Burr Type XII Failure Model,” Journal of Statistics Computation and Simulation, Vol. 44, 1992, pp. 31-40.

[23] E. K. AL-Hussaini and Z. F. Jaheen, “Approximate Bayes Estimators Applied to the Burr Model,” Communications in Statistics, Vol. 23, 1994, pp. 99-121.

[24] E. K. AL-Hussaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr XII Model,” Communications in Statistics, Vol. 24, 1995, pp. 1829-1842.

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

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

[27] N. Balakrishnan and M. Kateri, “Statistical Evidence in Contingency Tables Analysis,” Journal of Statistical Planning and Inference, Vol. 138, No. 4, 2008, pp. 873- 887. doi:10.1016/j.jspi.2007.02.005

[28] H. R. Varian, “A Bayesian Approach to Real State Assessment,” In: E. F. Stephen and A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage, North-Holland, Amsterdam, 1975, pp. 195-208.

[29] X. Y. Li, Y. Shi, J. Wei and J. Chai, “Empirical Bayes Estimators of Reliability Performances Using LINEX loss under Progressively Type-II Censored Samples,” Mathematics and Computers in Simulation, Vol. 73, No. 5, 2007, pp. 320-326. doi:10.1016/j.matcom.2006.05.002

[30] G. Prakash and D. C. Singh, “Shrinkage Estimation in Exponential Type-II Censored Data under LINEX Loss,” Journal of the Korean Statistical Society, Vol. 37, No. 1, 2008, pp. 53-61. doi:10.1016/j.jkss.2007.07.002

[31] D. K. Day, M. Ghosh and C. Srinivasan, “Simultaneous Estimation of Parameters under Entropy Loss,” Journal of Statistical Planning and Inference, Vol. 52, 1987, pp. 347-363. doi:10.1016/0378-3758(86)90108-4

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

[33] R. M. Soland, “Bayesian analysis of the Weibull Process with Unknown Scale and Shape Parameters,” IEEE Transactions on Reliability, Vol. 18, No. 4, 1969, pp. 181-184. doi:10.1109/TR.1969.5216348

[34] A. A. Soliman, A. H. Abd Ellah and K. S. Sultan, “Comparison of Estimates Using Record Statistics from Wei- bull Model: Bayesian and Non-Bayesian Approaches,” Computational Statistics & Data Analysis, Vol. 51, No. 3, 2006, pp. 2065-2077. doi:10.1016/j.csda.2005.12.020

[35] H. F. Martz and R. A. Waller, “Bayesian Reliability Ana- lysis,” Wiley, New York, 1982.

[36] D. M. Stablein, W. H. Carter and J. W. Novak, “Analysis of Survival Data with Non-Proportional Hazard Functions,” Controlled Clinical Trials, Vol. 2, No. 2, 1981, pp. 149-159. doi:10.1016/0197-2456(81)90005-2

[1] R. Viveros, N. Balakrishnan, “Interval Estimation of Pa- rameters of Life from Progressively Censored Data,” Te- chnometrics, Vol. 36, No. 1, 1994, pp. 84-91. doi:10.2307/1269201

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

[3] N. Balakrishnan and R. A. Sandhu, “Best Linear Unbiased and Maximum Likelihood Estimation for Exponential Distributions under General Progressive Type-II Censored Samples,” Sankhya: The Indian Journal of Seriestics, Vol. 58, No. 1, 1996, pp. 1-9.

[4] P. Mostert, J. Roux and A. Bekker, “Bayes Estimators of the Life Time Parameters Using the Compound Rayleigh model,” Journal of South African Statistical Association, Vol. 33, No. 2 ,1999, pp. 117-138.

[5] 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

[6] A. Bekker, J. Roux and P. Mostert, “A Generalization of the Compound Rayleigh Distribution: Using a Bayesian Methods on Cancer Survival Times,” Communications in Statistics—Theory and Methods, Vol. 29, No. 7, 2000, pp. 1419-1433. doi:10.1080/03610920008832554

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

[8] H. K. T. Ng, “Parameter estimation for a modifiied Wei- bull Distribution, for Progressively Type-II Censored Samples,” IEEE Transactions Reliability, Vol. 54, No. 3, 2005, pp. 374-380. doi:10.1109/TR.2005.853036

[9] N. Balakrishnan, N. Kannan, C. T. Lin and H. K. T. Ng, “Point and Interval Estimation 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

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

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

[12] D. G. Chen and Y. L. Lio, “Parameter Estimations for Generalized Exponential Distribution under Progressive Type-I Interval Censoring,” Computational Statistics and Data Analysis, Vol. 54, No. 6, 2010, pp. 1581-1591. doi:10.1016/j.csda.2010.01.007

[13] N. Balakrishnan and R. Aggarwala, “Progressive Censoring—Theory, Methods and Applications,” Birkh?user, Boston, 2000.

[14] N. Balakrishnan, “Progressive Censoring Methodology: An Appraisal,” Test, Vol. 16, No. 2007, pp. 211-289. doi:10.1007/s11749-007-0061-y

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

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

[17] 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

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

[19] J.-W. Wu, T.-R. Tsai and L.-Y. Ouyang, “Limited Failure-Censored Life Test for the Weibull Distribution,” IEEE Transactions on Reliability, Vol. 50, No. 1, 2001, pp. 107-111. doi:10.1109/24.935024

[20] W.-C. Lee, L.-W. Wu and H.-Y. Yu, “Statistical Inference about the Shape Parameter of the Bathtub-Shaped Distribution under the Failure-Censored Sampling Plan,” International Journal of Information and Management Sciences, Vol. 18, No. 2, 2007, pp. 157-172.

[21] 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. 1-12.

[22] E. K. AL-Hussaini and Z. F. Jaheen, “Bayes Estimation of the Parameters, Reliability and Failure Rate Functions of the Burr Type XII Failure Model,” Journal of Statistics Computation and Simulation, Vol. 44, 1992, pp. 31-40.

[23] E. K. AL-Hussaini and Z. F. Jaheen, “Approximate Bayes Estimators Applied to the Burr Model,” Communications in Statistics, Vol. 23, 1994, pp. 99-121.

[24] E. K. AL-Hussaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr XII Model,” Communications in Statistics, Vol. 24, 1995, pp. 1829-1842.

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

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

[27] N. Balakrishnan and M. Kateri, “Statistical Evidence in Contingency Tables Analysis,” Journal of Statistical Planning and Inference, Vol. 138, No. 4, 2008, pp. 873- 887. doi:10.1016/j.jspi.2007.02.005

[28] H. R. Varian, “A Bayesian Approach to Real State Assessment,” In: E. F. Stephen and A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage, North-Holland, Amsterdam, 1975, pp. 195-208.

[29] X. Y. Li, Y. Shi, J. Wei and J. Chai, “Empirical Bayes Estimators of Reliability Performances Using LINEX loss under Progressively Type-II Censored Samples,” Mathematics and Computers in Simulation, Vol. 73, No. 5, 2007, pp. 320-326. doi:10.1016/j.matcom.2006.05.002

[30] G. Prakash and D. C. Singh, “Shrinkage Estimation in Exponential Type-II Censored Data under LINEX Loss,” Journal of the Korean Statistical Society, Vol. 37, No. 1, 2008, pp. 53-61. doi:10.1016/j.jkss.2007.07.002

[31] D. K. Day, M. Ghosh and C. Srinivasan, “Simultaneous Estimation of Parameters under Entropy Loss,” Journal of Statistical Planning and Inference, Vol. 52, 1987, pp. 347-363. doi:10.1016/0378-3758(86)90108-4

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

[33] R. M. Soland, “Bayesian analysis of the Weibull Process with Unknown Scale and Shape Parameters,” IEEE Transactions on Reliability, Vol. 18, No. 4, 1969, pp. 181-184. doi:10.1109/TR.1969.5216348

[34] A. A. Soliman, A. H. Abd Ellah and K. S. Sultan, “Comparison of Estimates Using Record Statistics from Wei- bull Model: Bayesian and Non-Bayesian Approaches,” Computational Statistics & Data Analysis, Vol. 51, No. 3, 2006, pp. 2065-2077. doi:10.1016/j.csda.2005.12.020

[35] H. F. Martz and R. A. Waller, “Bayesian Reliability Ana- lysis,” Wiley, New York, 1982.

[36] D. M. Stablein, W. H. Carter and J. W. Novak, “Analysis of Survival Data with Non-Proportional Hazard Functions,” Controlled Clinical Trials, Vol. 2, No. 2, 1981, pp. 149-159. doi:10.1016/0197-2456(81)90005-2