Abstract
In the current paper, based on progressive type-II hybrid censored samples, the maximum likelihood and Bayes estimates for the two parameter Burr XII distribution are obtained. We propose the use of expectation-maximization (EM) algorithm to compute the maximum likelihood estimates (MLEs) of model parameters. Further, we derive the asymptotic variance-covariance matrix of the MLEs by applying the missing information principle and it can be utilized to construct asymptotic confidence intervals (CIs) for the parameters. The Bayes estimates of the unknown parameters are obtained under the assumption of gamma priors by using Lindley’s approximation and Markov chain Monte Carlo (MCMC) technique. Also, MCMC samples are used to construct the highest posterior density (HPD) credible intervals. Simulation study is conducted to investigate the accuracy of the estimates and compare the performance of CIs obtained. Finally, one real data set is analyzed for illustrative purposes.
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References
Abd-Elfattah AM, Hassan AS, Nassr SG (2008) Estimation in step-stress partially accelerated life tests for the Burr type XII distribution using type I censoring. Stat Methodol 5:502–514
Ahmed A, Soliman AH, Abd Ellah NA, Abou-Elheggag A, Modhesh A (2011) Bayesian inference and prediction of Burr type XII distribution for progressive first failure censored sampling. Intell Inform Manage 3:175–185
Ali Mousa MAM, Jaheen ZF (2002) Statistical inference for the Burr model based on progressively censored data. Computers & Mathematics with Applications 43:1441–1449
Banerjee A, Kundu D (2008) Inference based on type-II hybrid censored data from a Weibull distribution. IEEE Trans Reliab 57:369–378
Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8:69–92
Childs A, Chandrasekar B, Balakrishnan N (2008) Exact likelihood inference for an exponential parameter under progressive hybrid censoring schemes. In: Vonta F, Nikulin M, Limnios N, Huber-Carol C (eds) Statistical models and methods for biomedical and technical systems. Birkhäuser, pp 323–334
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J R Stat Soc Ser B 39:1–38
Epstein B (1954) Truncated life tests in the exponential case. Ann Math Stat 25:555–564
Gurunlu Alma O, Arabi Belaghi R (2015) On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data. J Stat Comput Simul 86(3):569–596. http://dx.doi.org/10.1080/00949655.2015.1025785
Gupta RD, Kundu D (2001) Exponentiated exponential family an alternative to Gamma and Weibull. Biometr J 33:117–130
Kundu D, Joarder A (2006) Analysis of type-II progressively hybrid censored data. Computational Statistics & Data Analysis 50:2509–2528
Kundu D, Pradhan B (2009) Estimating the parameters of the generalized exponential distribution in prescence of hybrid censoring. Communications in Statistics–Theory and Methods 38:2030–2041
Lindley DV (1980) Approximate bayesian method. Trabajos de Estadistica 31:223–237
Lin CT, Huang YL (2011) On progressive hybrid censored exponential distribution. J Stat Comput Simul. 1st published on: 21 June 2011 (iFirst)
Lin CT, Huang YL, Balakrishnan N (2011) Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring. J Stat Comput Simul 81:873–882
Louis TA (1982) Finding the observed information matrix using the EM algorithm. Journal of Royal Statistical Society: Series B 44:226–233
McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New York
Ng HKT, Chan PS, Balakrihnan N (2002) Estimation of parameters from progressively censored data using EM algorithm. Computational Statistics & Data Analysis 39:371–386
Rastogi MK, Tripathi YM (2013) Inference on unknown parameters of a Burr distribution under hybrid censoring. Stat Pap 54:619–643
Soliman AA (2005) Estimation of parameters of life from progressively censored data using Burr XII model. IEEE Trans Reliab 54:34–42
Varian HR (1975) A Bayesian approach to real estate assessment. In: Stephen EF, Zellner A (eds) Studies in bayesian econometrics and statistics in honor of Leonard J. Savage, North-Holland, pp 195–208
Wang F, Cheng YF (2010) EM algorithm for estimating the Burr XII parameters with multiple censored data. Qual Reliab Eng Int 26:615–630
Wingo DR (1993) Maximum likelihood methods for fitting the Burr type XII distribution to multiply (progressively) censored life test data. Metrika 40:203–210
Zellner A (1986) Bayesian estimation and prediction using asymptotic loss function. J Am Stat Assoc 81:446–451
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Noori Asl, M., Belaghi, R.A. & Bevrani, H. On Burr XII Distribution Analysis Under Progressive Type-II Hybrid Censored Data. Methodol Comput Appl Probab 19, 665–683 (2017). https://doi.org/10.1007/s11009-016-9514-7
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DOI: https://doi.org/10.1007/s11009-016-9514-7
Keywords
- Bayesian estimate
- EM algorithm
- Missing information principle
- Lindley’s approximation
- Importance sampling
- Progressive type-II hybrid censoring