Inference for the Chen Distribution Under Progressive First-Failure Censoring

  • Tanmay Kayal
  • Yogesh Mani TripathiEmail author
  • Liang Wang
Original Article


We consider estimation and prediction for the two-parameter Chen distribution on the basis of progressive first failure censoring. The classical estimates of parameters are obtained. In particular, the least square estimation is considered. Further Bayes estimates are derived using the Tierney and Kadane method. We have also used the MH algorithm for this purpose. Further asymptotic, Bonferroni and Bayesian intervals are constructed. The one- and two-sample prediction problems are discussed as well. We examine the performance of studied methods by performing simulations. We analyze a real data set for illustrative purposes, and some concluding remarks are also presented.


Bayes estimates HPD interval Least square estimation Maximum likelihood estimates Progressive first failure censoring 



Authors are thankful to a reviewer for very helpful comments which led to significant improvement in both content and presentation of the manuscript. Authors also thank the Editor for constructive suggestions. Yogesh Mani Tripathi gratefully acknowledges the partial financial support for this research work under a Grant EMR/2016/001401 SERB, India.


  1. 1.
    Balakrishnan N, Aggarwala R (2000) Progressive censoring: theory, methods, and applications. Springer, BerlinGoogle Scholar
  2. 2.
    Balakrishnan N, Kundu D (2013) Hybrid censoring: models, inferential results and applications. Comput Stat Data Anal 57(1):166–209MathSciNetzbMATHGoogle Scholar
  3. 3.
    Balasooriya U (1995) Failure-censored reliability sampling plans for the exponential distribution. J Stat Comput Simul 52(4):337–349zbMATHGoogle Scholar
  4. 4.
    Wu SJ, Kuş C (2009) On estimation based on progressive first-failure-censored sampling. Comput Stat Data Anal 53(10):3659–3670MathSciNetzbMATHGoogle Scholar
  5. 5.
    Asgharzadeh A (2006) Point and interval estimation for a generalized logistic distribution under progressive type II censoring. Commun Stat Theory Methods 35(9):1685–1702MathSciNetzbMATHGoogle Scholar
  6. 6.
    Balakrishnan N (2007) Progressive censoring methodology: an appraisal. Test 16(2):211MathSciNetzbMATHGoogle Scholar
  7. 7.
    Pradhan B, Kundu D (2009) On progressively censored generalized exponential distribution. Test 18(3):497MathSciNetzbMATHGoogle Scholar
  8. 8.
    Singh S, Tripathi YM, Wu SJ (2015) On estimating parameters of a progressively censored lognormal distribution. J Stat Comput Simul 85(6):1071–1089MathSciNetGoogle Scholar
  9. 9.
    Huang SR, Wu SJ (2012) Bayesian estimation and prediction for Weibull model with progressive censoring. J Stat Comput Simul 82(11):1607–1620MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rastogi MK, Tripathi YM, Wu SJ (2012) Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring. J Appl Stat 39(11):2389–2411MathSciNetGoogle Scholar
  11. 11.
    Wu SJ, Huang SR (2012) Progressively first-failure censored reliability sampling plans with cost constraint. Comput Stat Data Anal 56(6):2018–2030MathSciNetzbMATHGoogle Scholar
  12. 12.
    Potdar KG, Shirke DT (2014) Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data. J Stat Comput Simul 84(1):171–185MathSciNetGoogle Scholar
  13. 13.
    Singh S, Tripathi YM (2015) Reliability sampling plans for a lognormal distribution under progressive first-failure censoring with cost constraint. Stat Pap 56(3):773–817MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dube M, Krishna H, Garg R (2016) Generalized inverted exponential distribution under progressive first-failure censoring. J Stat Comput Simul 86(6):1095–1114MathSciNetGoogle Scholar
  15. 15.
    Chen Z (2000) A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Stat Probab Lett 49(2):155–161MathSciNetzbMATHGoogle Scholar
  16. 16.
    Aarset MV (1987) How to identify a bathtub hazard rate. IEEE Trans Reliab 36(1):106–108zbMATHGoogle Scholar
  17. 17.
    Wu SJ (2008) Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring. J Appl Stat 35(10):1139–1150MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ahmed EA (2014) Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach. J Appl Stat 41(4):752–768MathSciNetGoogle Scholar
  19. 19.
    Kayal T, Tripathi YM, Singh DP, Rastogi MK (2017) Estimation and prediction for Chen distribution with bathtub shape under progressive censoring. J Stat Comput Simul 87(2):348–366MathSciNetGoogle Scholar
  20. 20.
    Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (Methodological) 39:1–38MathSciNetzbMATHGoogle Scholar
  21. 21.
    Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B (Methodological) 44:226–233MathSciNetzbMATHGoogle Scholar
  22. 22.
    Swain JJ, Venkatraman S, Wilson JR (1988) Least-squares estimation of distribution functions in Johnson’s translation system. J Stat Comput Simul 29(4):271–297Google Scholar
  23. 23.
    Hossain A, Zimmer W (2003) Comparison of estimation methods for Weibull parameters: complete and censored samples. J Stat Comput Simul 73(2):145–153MathSciNetzbMATHGoogle Scholar
  24. 24.
    Helu A (2015) On the maximum likelihood and least squares estimation for the inverse Weibull parameters with progressively first-failure censoring. Open J Stat 5(01):75Google Scholar
  25. 25.
    Montgomery DC, Peck EA, Vining GG (2015) Introduction to linear regression analysis. Wiley, HobokenzbMATHGoogle Scholar
  26. 26.
    Kyriakides E, Heydt GT (2006) Calculating confidence intervals in parameter estimation: a case study. IEEE Trans Power Deliv 21(1):508–509Google Scholar
  27. 27.
    Kundu D, Pradhan B (2009) Bayesian inference and life testing plans for generalized exponential distribution. Sci China Ser A Math 52(6):1373–1388MathSciNetzbMATHGoogle Scholar
  28. 28.
    Lindley DV (1980) Approximate Bayesian methods. Trabajos de estadística y de investigación operativa 31(1):223–245MathSciNetzbMATHGoogle Scholar
  29. 29.
    Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81(393):82–86MathSciNetzbMATHGoogle Scholar
  30. 30.
    Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092Google Scholar
  31. 31.
    Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109MathSciNetzbMATHGoogle Scholar
  32. 32.
    Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8(1):69–92MathSciNetGoogle Scholar
  33. 33.
    Asgharzadeh A, Valiollahi R, Kundu D (2015) Prediction for future failures in Weibull distribution under hybrid censoring. J Stat Comput Simul 85(4):824–838MathSciNetzbMATHGoogle Scholar
  34. 34.
    Dey S, Singh S, Tripathi YM, Asgharzadeh A (2016) Estimation and prediction for a progressively censored generalized inverted exponential distribution. Stat Methodol 32:185–202MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hand DJ, Daly F, McConway K, Lunn D, Ostrowski E (1993) A handbook of small data sets, vol 1. CRC Press, Boca RatonzbMATHGoogle Scholar
  36. 36.
    Bain LJ (1974) Analysis for the linear failure-rate life-testing distribution. Technometrics 16(4):551–559MathSciNetzbMATHGoogle Scholar
  37. 37.
    Gupta A, Rohatgi V (1980) On the estimation of restricted mean. J Stati Plann Inference 4(4):369–379MathSciNetzbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology PatnaBihtaIndia
  2. 2.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China

Personalised recommendations