Estimation and Prediction for a Progressively First-Failure Censored Inverted Exponentiated Rayleigh Distribution

  • Raj Kamal Maurya
  • Yogesh Mani TripathiEmail author
  • Manoj Kumar Rastogi
Original Article


We discuss inverted exponentiated Rayleigh distribution under progressive first-failure censoring. Maximum likelihood and Bayes estimates of unknown parameters are obtained. An expectation–maximization algorithm is used for computing maximum likelihood estimates. Asymptotic intervals are constructed from the observed Fisher information matrix. Bayes estimates of unknown parameters are obtained under the squared error loss function. We construct highest posterior density intervals based on importance sampling. Different predictors and prediction intervals of censored observations are discussed. A Monte Carlo simulations study is performed to compare different methods. Finally, three real data sets are analyzed for illustration purposes.


Progressive first-failure censoring Expectation–maximization algorithm Lindley approximation Tirney and Kadane method Importance sampling method HPD intervals 

Mathematics Subject Classification

62N01 62N02 62N05 



The authors are grateful to a reviewer for encouraging comments and constructive suggestions that led to significant improvement in presentation and content of the manuscript. They also thank the Editor for helpful comments. Yogesh Mani Tripathi gratefully acknowledges the partial financial support for this research work under a Grant EMR/2016/001401 Science and Engineering Research Board, India.


  1. 1.
    Ahmadi MV, Doostparast M, Ahmadi J (2013) Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme. Journal of Computational and Applied Mathematics 239:93–102MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balakrishnan N, Aggarwala R (2000) Progressive censoring: theory, methods, and applications. Springer, BerlinCrossRefGoogle Scholar
  3. 3.
    Balakrishnan N, Beutner E, Kateri M (2009) Order restricted inference for exponential step-stress models. IEEE Trans Reliab 58(1):132–142CrossRefGoogle Scholar
  4. 4.
    Balasooriya U (1995) Failure-censored reliability sampling plans for the exponential distribution. J Stat Comput Simul 52(4):337–349CrossRefGoogle Scholar
  5. 5.
    Chen MH, Shao QM (1999) Monte carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8(1):69–92MathSciNetGoogle Scholar
  6. 6.
    Chen Z (1997) Parameter estimation of the Gompertz population. Biom J 39(1):117–124MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chhikara R, Folks J (1977) The inverse gaussian distribution as a lifetime model. Technometrics 19(4):461–468CrossRefGoogle Scholar
  8. 8.
    Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (Methodol) 39:1–22MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dey S, Dey T (2014) On progressively censored generalized inverted exponential distribution. J Appl Stat 41(12):2557–2576MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dube M, Krishna H, Garg R (2016) Generalized inverted exponential distribution under progressive first-failure censoring. J Stat Comput Simul 86(6):1095–1114MathSciNetCrossRefGoogle Scholar
  11. 11.
    Efron B (1988) Logistic regression, survival analysis, and the Kaplan–Meier curve. J Am Stat Assoc 83(402):414–425MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ghitany M, Al-Jarallah R, Balakrishnan N (2013) On the existence and uniqueness of the mles of the parameters of a general class of exponentiated distributions. Statistics 47(3):605–612MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghitany M, Tuan V, 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
  14. 14.
    Gibbons DI, Vance LC (1983) Estimators for the 2-parameter Weibull distribution with progressively censored samples. IEEE Trans Reliab 32(1):95–99CrossRefGoogle Scholar
  15. 15.
    Hand DJ, Daly F, McConway K, Lunn D, Ostrowski E (1993) A handbook of small data sets, vol 1. CRC Press, Boca RatonzbMATHGoogle Scholar
  16. 16.
    Huang SR, Wu SJ (2012) Bayesian estimation and prediction for Weibull model with progressive censoring. J Stat Comput Simul 82(11):1607–1620MathSciNetCrossRefGoogle Scholar
  17. 17.
    Johnson LG (1964) Theory and technique of variation research. Elsevier Publishing Company, AmsterdamzbMATHGoogle Scholar
  18. 18.
    Kaushik A, Pandey A, Maurya SK, Singh U, Singh SK (2017) Estimations of the parameters of generalised exponential distribution under progressive interval type-I censoring scheme with random removals. Austrian J Stat 46(2):33–47CrossRefGoogle 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):348–366MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kızılaslan F (2018) Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on a general class of inverse exponentiated distributions. Stat Pap 59(3):1161–1192MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kundu D, Pradhan B (2009) Bayesian inference and life testing plans for generalized exponential distribution. Sci China Ser A Math 52(6):1373–1388MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lindley DV (1980) Approximate Bayesian methods methods. Trabajos de estadística y de investigación operativa 31(1):223–245MathSciNetCrossRefGoogle Scholar
  23. 23.
    Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B (Methodol) 44:226–233MathSciNetzbMATHGoogle Scholar
  24. 24.
    Madi M, Raqab M (2009) Bayesian analysis for the exponentiated Rayleigh distribution. Metron Int J Stat 67:269–288zbMATHGoogle Scholar
  25. 25.
    Mann NR (1971) Best linear invariant estimation for weibull parameters under progressive censoring. Technometrics 13(3):521–533MathSciNetCrossRefGoogle Scholar
  26. 26.
    Maurya RK, Tripathi YM, Rastogi MK, Asgharzadeh A (2017) Parameter estimation for a Burr type XII distribution under progressive censoring. Am J Math Manag Sci 36(3):259–276Google Scholar
  27. 27.
    Ng H, Chan P, Balakrishnan N (2002) Estimation of parameters from progressively censored data using EM algorithm. Comput Stat Data Anal 39(4):371–386MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pradhan B, Kundu D (2009) On progressively censored generalized exponential distribution. Test 18(3):497–515MathSciNetCrossRefGoogle Scholar
  29. 29.
    Raqab MZ, Madi MT (2011) Inference for the generalized Rayleigh distribution based on progressively censored data. J Stat Plan Inference 141(10):3313–3322MathSciNetCrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Soliman AA (2005) Estimation of parameters of life from progressively censored data using Burr-XII model. IEEE Trans Reliab 54(1):34–42CrossRefGoogle Scholar
  32. 32.
    Soliman AA, Abd-Ellah AH, Abou-Elheggag NA, Abd-Elmougod GA (2012) Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data. Comput Stat Data Anal 56(8):2471–2485MathSciNetCrossRefGoogle Scholar
  33. 33.
    Soliman AA, Ellah AA, Abou-Elheggag NA, Modhesh A (2011) Bayesian inference and prediction of Burr type XII distribution for progressive first failure censored sampling. Intell Inf Manag 3(05):175zbMATHGoogle Scholar
  34. 34.
    Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81(393):82–86MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wu SJ, Chen DH, Chen ST (2006) Bayesian inference for Rayleigh distribution under progressive censored sample. Appl Stoch Models Business Ind 22(3):269–279MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wu SJ, Kuş C (2009) On estimation based on progressive first-failure-censored sampling. Comput Stat Data Anal 53(10):3659–3670MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2019

Authors and Affiliations

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

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