Bayesian analysis of heterogeneous doubly censored lifetime data using the 3-component mixture of Rayleigh distributions: A Monte Carlo simulation study

Document Type : Article


1 Department of Statistics, Government College University, Faisalabad 38000, Pakistan

2 Department of Mathematics and Statistics, Riphah International University, Islamabad 44000, Pakistan.

3 Department of Statistics, Quaid-i-Azam University, Islamabad 44000, Pakistan.

4 Department of Statistics, Government College University, Faisalabad 38000, Pakistan.


This article is about Bayesian estimation of parameters of a heterogeneous 3-component mixture of Rayleigh distributions (3-CMRD) generating a mixture data. Being the most popular and reasonable sampling scheme in reliability and survival analyses, the doubly censored sampling scheme is considered. The Bayes estimators and their posterior risks are derived under various situations. In addition, elicitation of hyperparameters is presented. Algebraic expressions for posterior predictive distribution and Bayesian predictive intervals are derived.  Assuming the informative and the non-informative priors, a comprehensive Monte Carlo simulation is conducted to examine the performance of the Bayes estimators under symmetric and asymmetric loss functions. Finally, to highlight the practical importance, the proposed 3-compnent mixture model is applied to a doubly censored lifetime data from a real life situation. It is observed that the analysis of doubly censored data in Bayesian framework, the SRIGP paired with SELF (DLF) is suitable choice for estimating mixing proportion (component) parameters.


Main Subjects

1. Fernandez, A.J. "On maximum likelihood prediction based on type-II doubly censored exponential data", Metrika, 50, pp. 211-220 (2000).
2. Khan, H.M.R., Provost, S.B., and Singh, A. "Predictive inference from a two-parameter Rayleigh life model given a doubly censored sample", Communications in Statistics-Theory and Methods, 39, pp. 1237- 1246 (2010).
3. Kim, C. and Song, S. "Bayesian estimation of the parameters of the generalized exponential distribution from doubly censored samples", Statistical Papers, 51, pp. 583-597 (2010).
4. Khan, H.M.R., Albatineh, A., Alshahrani, S., Jenkins, N., and Ahmed, N.U. "Sensitivity analysis of predictive modeling for responses from the three-parameter Weibull model with a follow-up doubly censored sample of cancer patients", Computational Statistics and Data Analysis, 55, pp. 3093-3103 (2011).
5. Pak, A., Parham, G.A., and Saraj, M. "On estimation of Rayleigh scale parameter under doubly type-II censoring from imprecise data", Journal of Data Science, 11, pp. 305-322 (2013).
6. Feroze, N. and Aslam, M. "Bayesian analysis of doubly censored lifetime data using two-component mixture of Weibull distribution", Journal of the National Science Foundation of Sri Lanka, 42(4), pp. 325-334 (2014).
7. Sindhu, T.N., Feroze, N., and Aslam, M. "Analysis of doubly censored Burr type-II distribution: a Bayesian look", Electronic Journal of Applied Statistical Analysis, 8(2), pp. 154-169 (2015).
8. Rattanapitikon, W. "Verification of conversion formulas for computing representative wave heights", Ocean Engineering, 37, pp. 1554-1563 (2010).
9. Siddiqui, M.M. "Some problems connected with Rayleigh distributions", The Journal of Research of the National Bureau of Standards, 60(D), pp. 167-174 (1962).
10. Ahmed, S.A. and Mahammed, H.O. "A statistical analysis of wind power density based on Weibull and Rayleigh models of "Penjwen Region" Sulaimani/ Iraq", Jordan Journal of Mechanical and Industrial Engineering, 6(2), pp. 135-140 (2012).
11. Chivers, R.C. "The scattering of ultrasound by human tissues, some theoretical models", Ultrasound Medical Biology, 3, pp. 1-13 (1977).
12. Ali, S., Aslam, M., Kundu, D., and Kazmi, S.M.A. "Bayesian estimation of the mixture of generalized exponential distribution: a versatile lifetime model in industrial processes", Journal of the Chinese Institute of Industrial Engineers, 29(4), pp. 246-269 (2012).
13. Bhattacharya, C.G. "A simple method of resolution of a distribution into Gaussian components", Biometrics, 23, pp. 115-135 (1967).
14. Harris, C.M. "On finite mixtures of geometric and negative binomial distributions", Communications in Statistics-Theory and Methods, 12, pp. 987-1007 (1983).
15. Jedidi, K., Jagpal, H.S., and DeSarbo, W.S. "Finitemixture structural equation models for response-based segmentation and unobserved heterogeneity", Marketing Science, 16(1), pp. 39-59 (1997).
16. Shawky, A.I. and Bakoban, R.A. "On finite mixture of two-component Exponentiated Gamma distribution", Journal of Applied Sciences Research, 5(10), pp. 1351- 1369 (2009).
17. Sultan, K.S., Ismail, M.A., and Al-Moisheer, A.S. "Mixture of two inverse Weibull distributions: Properties and estimation", Computational Statistics & Data Analysis, 51(1), pp. 5377-5387 (2007).
18. Santos, A.M. "Robust estimation of censored mixture models", PhD Thesis, University of Colorado Denver (2011).
19. Al-Hussaini, E.K. and Hussein, M. "Estimation under a finite mixture of exponentiated exponential components model and balanced square error loss", Open Journal of Statistics, 2, pp. 28-38 (2012).
20. Mohammadi, A. and Salehi-Rad, M.R. "Bayesian inference and prediction in an M/G/1 with optional second service", Communications in Statistics-Simulation and Computation, 41(3), pp. 419-435 (2012).
21. Ahmad, A.E.A. and Al-Zaydi, A.M. "Inferences under a class of finite mixture distributions based on generalized order statistics", Open Journal of Statistics, 3, pp. 231-244 (2013).
22. Mohammadi, A., Salehi-Rad, M.R., and Wit, E.C. "Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service", Computational Statistics, 28(2), pp. 683-700 (2013).
23. Ali, S. "Mixture of the inverse Rayleigh distribution: properties and estimation in Bayesian framework", Applied Mathematical Modelling, 39(2), pp. 515-530 (2014).
24. Ateya, S.F. "Maximum likelihood estimation under a finite mixture of generalized exponential distributions based on censored data", Statistical Papers, 55(2), pp. 311-325 (2014).
25. Mohamed, M.M., Saleh, E., and Helmy, S.M. "Bayesian prediction under a finite mixture of generalized Exponential lifetime model", Pakistan Journal of Statistics and Operation Research, 10(4), pp. 417-433 (2014).
26. Zhang, H. and Huang, Y. "Finite mixture models and their applications: a review", Austin Biometrics and Biostatistics, 2(1), pp. 1-6 (2015).
27. Aslam, M. "An application of prior predictive distribution to elicit the prior density", Journal of Statistical Theory and Applications, 2, pp. 70-83 (2003).
28. Legendre, A.M. "New methods for the determination of cometary orbits: Appendix on the least squares method" [Nouvelles methodes pour la determination des orbites des cometes: Appendice sur la methode des moindres carres], Gautheir-Villars, Paris (1806).
29. Norstrom, J.G. "The use of precautionary loss function in risk analysis", Reliability, IEEE Transactions on, 45(3), pp. 400-403 (1996).
30. DeGroot. M.H., Optimal Statistical Decision, McGraw-Hill (2005).
31. Gomez, Y.M., Bolfarine, H., and Gomez, H.W. "A new extension of the Exponential distribution", Revista Colombiana de Estadistica, 37(1), pp. 25-34 (2014).