Inference for an Inverted Exponentiated Pareto Distribution Under Progressive Censoring

  • Raj Kamal Maurya
  • Yogesh Mani Tripathi
  • Tanmay Sen
  • Manoj Kumar RastogiEmail author
Original Article


In this paper, estimation of unknown parameters of an inverted exponentiated Pareto distribution is considered under progressive Type-II censoring. Maximum likelihood estimates are obtained from the expectation–maximization algorithm. We also compute the observed Fisher information matrix. In the sequel, asymptotic and bootstrap-p intervals are constructed. Bayes estimates are derived using the importance sampling procedure with respect to symmetric and asymmetric loss functions. Highest posterior density intervals of unknown parameters are constructed as well. The problem of one- and two-sample prediction is discussed in Bayesian framework. Optimal plans are obtained with respect to two information measure criteria. We assess the behavior of suggested estimation and prediction methods using a simulation study. A real dataset is also analyzed for illustration purposes. Finally, we present some concluding remarks.


Expectation–maximization algorithm Bootstrap interval Importance sampling method HPD interval Bayes prediction Optimal censoring 



The authors are thankful to the reviewers for their valuable suggestions which have significantly improved the content and the presentation of our paper. They also thank the Editor and an Associate Editor for the encouraging comments. Yogesh Mani Tripathi gratefully acknowledges the partial financial support for this research work under a Grant EMR/2016/001401 SERB, India.


  1. 1.
    Abouammoh AM, Alshingiti AM (2009) Reliability estimation of generalized inverted exponential distribution. J Stat Comput Simul 79(11):1301–1315MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asgharzadeh A (2006) Point and interval estimation for a generalized logistic distribution under progressive type II censoring. Commun Stat Theory Methods 35(9):1685–1702MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Balakrishnan N, Aggarwala R (2000) Progressive censoring: theory, methods, and applications. Springer Science and Business Media, BerlinCrossRefGoogle Scholar
  4. 4.
    Balakrishnan N (2007) Progressive censoring methodology: an appraisal. Test 16(2):211–259MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Balakrishnan N, Rad AH, Arghami NR (2007) Testing exponentiality based on Kullback–Leibler information with progressively Type-II censored data. IEEE Trans Reliab 56(2):301–307CrossRefGoogle Scholar
  6. 6.
    Balakrishnan N, Cramer E (2014) The art of progressive censoring: applications to reliability and quality. Birkhauser, New YorkzbMATHCrossRefGoogle Scholar
  7. 7.
    Bhattacharya R, Pradhan B, Dewanji A (2016) On optimum life-testing plans under Type-II progressive censoring scheme using variable neighborhood search algorithm. Test 25(2):309–330MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8(1):69–92MathSciNetGoogle Scholar
  9. 9.
    Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B 39:1–38MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dey S, Singh S, Tripathi YM, Asgharzadeh A (2016) Estimation and prediction for a progressively censored generalized inverted exponential distribution. Stat Methodol 32(1):185–202MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ghafoori S, Habibi Rad A, Doostparast M (2011) Bayesian two-sample prediction with progressively Type-II censored data for some lifetime models. J Iran Stat Soc 10(1):63–86MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ghitany ME, Tuan VK, Balakrishnan N (2014) Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data. J Stat Comput Simul 84(1):96–106MathSciNetCrossRefGoogle Scholar
  13. 13.
    Huang SR, Wu SJ (2012) Bayesian estimation and prediction for Weibull model with progressive censoring. J Stat Comput Simul 82(11):1607–1620MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, vol 2. Wiley, New YorkGoogle Scholar
  15. 15.
    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):348–366MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kohansal A (2018) On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample. Stat Pap. CrossRefGoogle Scholar
  17. 17.
    Kundu D (2008) Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics 50(2):144–154MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kundu D, Pradhan B (2009) Bayesian inference and life testing plans for generalized exponential distribution. Sci China Ser A Math 52(6):1373–1388MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B 44:226–233MathSciNetzbMATHGoogle Scholar
  20. 20.
    Maurya RK, Tripathi YM, Rastogi MK, Asgharzadeh A (2017) Parameter estimation for a Burr XII distribution under progressive censoring. Am J Math Manag Sci 36(3):259–276Google Scholar
  21. 21.
    Murthy DP, Xie M, Jiang R (2004) Weibull models. Wiley, New York, p 505zbMATHGoogle Scholar
  22. 22.
    Ng HKT, Chan PS, Balakrishnan N (2004) Optimal progressive censoring plans for the Weibull distribution. Technometrics 46(4):470–481MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nigm AM, Al-Hussaini EK, Jaheen ZF (2003) Bayesian one-sample prediction of future observations under Pareto distribution. Statistics 37(6):527–536MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Pradhan B, Kundu D (2009) On progressively censored generalized exponential distribution. Test 18(3):497–515MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ramos MW, Marinho PR, Silva RV, Cordeiro GM (2013) The exponentiated Lomax Poisson distribution with an application to lifetime data. Adv Appl Stat 34(2):107–135MathSciNetzbMATHGoogle Scholar
  26. 26.
    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):2389–2411MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rastogi MK, Tripathi YM (2014) Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring. J Appl Stat 41(11):2375–2405MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sinha SK (1998) Bayesian estimation. New Age International (P) Limited, New DelhiGoogle Scholar
  29. 29.
    Singh S, Tripathi YM, Wu SJ (2015) On estimating parameters of a progressively censored lognormal distribution. J Stat Comput Simul 85(6):1071–1089MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623-656 (Mathematical Reviews (MathSciNet): MR10, 133e)Google Scholar
  31. 31.
    Tanner MA (1991) Tools for statistical inference, vol 3. Springer, New YorkzbMATHCrossRefGoogle Scholar
  32. 32.
    Wang Z, Desmond AF, Lu X (2006) Modified censored moment estimation for the two-parameter Birnbaum–Saunders distribution. Comput Stat Data Anal 50(4):1033–1051MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Zellner A (1986) Bayesian estimation and prediction using asymmetric loss functions. J Am Stat Assoc 81(394):446–451MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Zheng G, Park S (2004) On the Fisher information in multiply censored and progressively censored data. Commun Stat Theory Methods 33(8):1821–1835MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2018

Authors and Affiliations

  • Raj Kamal Maurya
    • 1
  • Yogesh Mani Tripathi
    • 1
  • Tanmay Sen
    • 1
  • Manoj Kumar Rastogi
    • 2
    Email author
  1. 1.Department of MathematicsIndian Institute of Technology PatnaBihtaIndia
  2. 2.Department of StatisticsPatna UniversityPatnaIndia

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