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Statistical Papers

, Volume 60, Issue 3, pp 761–803 | Cite as

Estimation based on progressively type-I hybrid censored data from the Burr XII distribution

  • R. Arabi BelaghiEmail author
  • M. Noori Asl
Regular Article

Abstract

This study considers the problem of estimating unknown parameters of the Burr XII distribution under classical and Bayesian frameworks when samples are observed in the presence of progressively type-I hybrid censoring. Under classical approach, we employ EM and stochastic EM algorithm for obtaining the maximum likelihood estimators of model parameters. On the other hand, under Bayesian framework, we obtain Bayes estimators with respect to different symmetric and asymmetric loss functions under non-informative and informative priors. In this regard, we use Tierney–Kadane and importance sampling methods. Asymptotic normality theory and MCMC samples are employed to construct the confidence intervals and HPD credible intervals. To improve the estimation accuracy shrinkage pre-test estimation strategy is also suggested. The relative efficiency of these estimators with respect to both classical and Bayesian estimators are investigated numerically. Our simulation studies reveal that the shrinkage pre-test estimation strategy outperforms the estimation based on classical and Bayesian procedure. Finally, one real data set is analyzed to illustrate the methods of inference discussed here.

Keywords

Bayesian estimation SEM algorithm Importance sampling Tierney–Kadane method Shrinkage pre-test estimation Progressively type-I hybrid censored data 

Notes

Acknowledgements

The authors are thankful the reviewers for their valuable comments and suggestions which significantly improved the earlier version of this paper.

References

  1. Abdel-Hamid HH (2009) Constant-partially accelerated life tests for burr type-XII distribution with progressive type-II censoring. Comput Stat Data Anal 53:2511–2523MathSciNetCrossRefzbMATHGoogle Scholar
  2. Abuzaid AL (2015) The estimation of the Burr-XII parameters with middle-censored data. SpringerPlus.  https://doi.org/10.1186/s40064-015-0856-3
  3. Ahmed SE (1992) Shrinkage preliminary test estimation in multivariate normal distributions. J Stat Comput Simul 43:177–195MathSciNetCrossRefGoogle Scholar
  4. Al-Hussaini EK, Jaheen ZF (1992) Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model. J Stat Comput Simul 41:31–40MathSciNetCrossRefzbMATHGoogle Scholar
  5. Arabi Belaghi R, Arashi M, Tabatabaey SMM (2015) On the construction of preliminary test estimator based on record values for Burr 12 model. Commun Stat Theory Methods 44:1–23CrossRefzbMATHGoogle Scholar
  6. Bancroft TA (1944) On biases in estimation due to the use of preliminary tests of significances. Ann Math Stat 15:190–204MathSciNetCrossRefzbMATHGoogle Scholar
  7. Burr IW, Cislak PJ (1968) On a general system of distributions: 1. it’s curve-shaped characteristics; 2. the sample median. J Am Stat Assoc 63:627–635Google Scholar
  8. Calabria R, Pulcini G (1996) Point estimation under asymmetric loss functions for left-truncated exponential samples. Commun Stat Theory Methods 25:585–600MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Gr Stat 8:69–92MathSciNetGoogle Scholar
  10. Chien TL, Yen LH, Balakrishnan N (2011) Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring. J Stat Comput Simul 81:873–882MathSciNetCrossRefzbMATHGoogle Scholar
  11. Cramer E, Balakrishnan N (2013) On some exact distributional results based on type-I progressively hybrid censored data from exponential distributions. Stat Methodol 10:128–150MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the em algorithm. J R Stat Soc Ser B 39(1):1–38Google Scholar
  13. Diebolt J, Celeux G (1993) Asymptotic properties of a stochastic em algorithm for estimating mixing proportions. Stoch Models 9:599–613MathSciNetCrossRefzbMATHGoogle Scholar
  14. Epstein B (1954) Truncated life tests in the exponential case. Ann Math Stat 25:555–564MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kibria BMG (2004) Performance of the shrinkage preliminary tests ridge regression estimators based on the conflicting of W, LR and LM tests. J Stat Comput Simul 74:793–810MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kibria BMG, Saleh AKME (2004) Preliminary test ridge regression estimatorswith Student’s t errors and conflicting test-statistics. Metrika 59:105124CrossRefGoogle Scholar
  17. Kibria BMG, Saleh AKME (2005) Comparison between Han-Bancroft and Brook method to determine the optimum significance level for pre-test estimator. J Prob Stat Sci 3:293–303Google Scholar
  18. Kibria BMG, Saleh AKME (2006) Optimum critical value for pre-test estimators. Commun Stat Theory Methods 35:309–320MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kibria BMG, Saleh AKMD (2010) Preliminary test estimation of the parameters of exponential and Pareto distributions for censored samples. Stat Pap 51:757–773MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kundu D, Joarder A (2006) Analysis of type-II progressively hybrid censored data. Comput Stat Data Anal 50:2509–2528MathSciNetCrossRefzbMATHGoogle Scholar
  21. Li J, Ma L (2015) Inference for the generalized Rayleigh distribution based on progressively type-II hybrid censored data. J Inf Comput Sci 12:1101–1112CrossRefGoogle Scholar
  22. Lomax KS (1954) Business failure: another example of the analysis of failure data. J Am Stat Assoc 49:847–852CrossRefzbMATHGoogle Scholar
  23. Louis TA (1982) Finding the observed information matrix using the EM algorithm. J R Stat Soc Ser B 44:226–233MathSciNetzbMATHGoogle Scholar
  24. Mousa MA, Jaheen Z (2002a) Bayesian prediction for progressively censored data from the burr model. Stat Pap 43:587–593MathSciNetCrossRefzbMATHGoogle Scholar
  25. Mousa MA, Jaheen Z (2002b) Statistical inference for the burr model based on progressively censored data. Comput Math Appl 43:1441–1449MathSciNetCrossRefzbMATHGoogle Scholar
  26. Nelson W (1982) Applied life data analysis. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  27. Rastogi MK, Tripathi YM (2012) Estimating the parameters of a Burr distribution under progressive type II censoring. Stat Methodol 9:381–391MathSciNetCrossRefzbMATHGoogle Scholar
  28. Rastogi MK, Tripathi YM (2013) Inference on unknown parameters of a Burr distribution under hybrid censoring. Stat Pap 54:619–643MathSciNetCrossRefzbMATHGoogle Scholar
  29. Rodriguez RN (1977) A guide to the Burr type XII distributions. Biometrika 64:129–134MathSciNetCrossRefzbMATHGoogle Scholar
  30. Saleh AKME, Kibria BMG (1993) Performance of some new preliminary test ridge regression estimators and their properties. Commun Stat Theory Methods 22:2747–2764MathSciNetCrossRefzbMATHGoogle Scholar
  31. Saleh AKME, Sen PK (1978) Nonparametric estimation of location parameter after a preliminary test on regression. Ann Stat 6:154–168MathSciNetCrossRefzbMATHGoogle Scholar
  32. Singh S, Tripathi YM (2016) Estimating the parameters of an inverse weibull distribution under progressive type-I interval censoring. Stat Pap.  10.1007/s00362-016-0750-2
  33. Soliman A (2005) Estimation of parameters of life from progressively censored data using Burr XII model. IEEE Trans Reliab 54:34–42CrossRefGoogle Scholar
  34. Tadikamalla PR (1980) A look at the Burr and related distributions. Int Stat Rev 48:337–344MathSciNetCrossRefzbMATHGoogle Scholar
  35. Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81:82–86MathSciNetCrossRefzbMATHGoogle Scholar
  36. Varian HR (1975) A bayesian approach to real estate assessment. Stud Bayesian Econom Stat Honor Leonard J Savage, pp 195–208Google Scholar
  37. Wang FK, Cheng Y (2010) EM algorithm for estimating the Burr XII parameters with multiple censored data. Qual Reliab Eng Int 26:615–630CrossRefGoogle Scholar
  38. Wei GC, Tanner MA (1990) A monte carlo implementation of the em algorithm and the poor man’s data augmentation algorithms. J Am Stat Assoc 85:699–704CrossRefGoogle Scholar
  39. Wingo DR (1993) Maximum likelihood estimation of Burr XII distribution parameters under type II censoring. Microelectron Reliab 33:1251–1257CrossRefGoogle Scholar
  40. Zimmer WJ, Keats JB, Wang FK (1998) The Burr XII distribution in reliability analysis. J Qual Technol 30:386–394CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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