Estimation of exponentiated Nadarajah-Haghighi distribution under progressively type-II censored sample with application to bladder cancer data

Abstract

In this article we consider statistical inferences about the unknown parameters of the exponentiated Nadarajah-Haghighi (ENH) distribution based on progressively type-II censoring using classical and Bayesian procedures. For classical procedures, maximum likelihood (ML) and least square estimators of the unknown parameters are derived. The Bayes estimators are obtained based on both the symmetric (squared error) and asymmetric (LINEX, general entropy) loss functions. Furthermore, Markov chain Monte Carlo (MCMC) technique is used to compute the Bayes estimators and the associated credible intervals. Moreover, asymptotic confidence intervals are constructed using the normality property of the ML estimates. After that, the asymptotic confidence intervals using ML estimates and two parametric bootstrap confidence intervals are provided to compare with Bayes credible intervals. Finally, simulation study and a bladder cancer application are presented to illustrate the proposed methods of estimation.

References

  1. 1.

    E. A. Ahmed, Bayesian estimation based on progressive Type-II censoring from two-parameter bathtubshaped lifetime model: an Markov chain Monte Carlo approach, J. Appl. Stat., 41 (2014), 752–768.

    MathSciNet  Article  Google Scholar 

  2. 2.

    E. A. Ahmed, Estimation of some lifetime parameters of generalized Gompertz distribution under progressively type-II censored data, Appl. Math. Modell, 39 (2015), 5567–5578.

    MathSciNet  Article  Google Scholar 

  3. 3.

    N. Balakrishnan, Progressive censoring methodology: an appraisal (with discussions), Test, 16 (2007) 211–296

    MathSciNet  Article  Google Scholar 

  4. 4.

    N. Balakrishnan and R. Aggarwala, Progressive censoring: theory, methods, and applications, Boston: Birkhauser. (2000).

    Google Scholar 

  5. 5.

    N. Balakrishnan and E. Cramer, The arts of progressive censoring, Birkhauser: New York, NY, USA (2014).

    Google Scholar 

  6. 6.

    N. Balakrishnan and R. A. Sandhu, A simple simulational algorithm for generating progressive Type-II censored samples, Amer. Statist., 49 (1995), 229–230.

    Google Scholar 

  7. 7.

    J. O. Berger, Statistical decision theory and bayesian analysis, Springer, New York, (1985).

    Google Scholar 

  8. 8.

    R. Calabria and G. Pulcini, An engineering approach to Bayes estimation for the Weibull distribution, Microelectronics Reliability, 34 (1994), 789–802

    Article  Google Scholar 

  9. 9.

    B. Efron, Jackknife - after - bootstrap standard errors and infl uence functions, J. R. Stat. Soc. B, 54 (1992), 83–127.

    MATH  Google Scholar 

  10. 10.

    A. El-Gohary, A. Alshamrani, and A. Al-Otaibi, The generalized Gompertz distribution, Applied Mathematical Modeling, 37 (2013), 13–24.

    MathSciNet  Article  Google Scholar 

  11. 11.

    A. E. Gelfand and A. F. M. Smith, Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, 85 (1990), 398–409.

    MathSciNet  Article  Google Scholar 

  12. 12.

    S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6 (1984), 721–741.

    MATH  Google Scholar 

  13. 13.

    R. D. Gupta and D. Kundu, Generalized exponential distributions, Australian and New Zealand Journal of Statistics, 41 (1999), 173–188.

    MathSciNet  Article  Google Scholar 

  14. 14.

    R. D. Gupta and D. Kundu, Exponentiated exponential family: An alternative to gamma and weibull distributions, Biometrical Journal, 43 (2001), 117–130.

    MathSciNet  Article  Google Scholar 

  15. 15.

    P. Hall, The bootstrap and Edgeworth expansion, Springer-Verlag, New York, (1992).

    Google Scholar 

  16. 16.

    U. Hjorth, A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates, In: Technometrics, 22 (1980), 99–109.

    MATH  Google Scholar 

  17. 17.

    C. D. Lai, M. Xie, and D. N. P. Murthy, A modified Weibull distribution, IEEE Transactions on Reliability, 52 (2003), 33–37.

    Article  Google Scholar 

  18. 18.

    E. T. Lee and J. W. Wang, Statistical methods for survival data analysis (3rd ed.), New York: Wiley, (2003).

    Google Scholar 

  19. 19.

    A. J. Lemonte, A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function, Computational Statistics and Data Analysis, 62 (2013), 149–170.

    MATH  Google Scholar 

  20. 20.

    A. J. Lemonte and G. M. Cordeiro, The exponentiated generalized inverse Gaussian distribution, Statistics and Probability Letters, 81 (2011), 506–517.

    MathSciNet  Article  Google Scholar 

  21. 21.

    A. W. Marshall and I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84 (1997), 641–652.

    MathSciNet  Article  Google Scholar 

  22. 22.

    G. S. Mudholkar and D. K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure-real data, IEEE Transaction Reliability, 42 (1993), 299–302.

    Article  Google Scholar 

  23. 23.

    S. Nadarajah and F. Haghighi, An extension of the exponential distribution, Statistics, 45 (2011), 543–558.

    MathSciNet  Article  Google Scholar 

  24. 24.

    S. Rajarshi and M. B. Rajarshi, Bathtub distributions: A review, Communication in Statistics - Theory Methods, 17 (1988), 2597–2621.

    MathSciNet  Article  Google Scholar 

  25. 25.

    G. O. Silva, E. M. M. Ortega, and G. M. Cordeiro, The beta modifiedWeibull distribution, Lifetime Data Anaysis, 16 (2010), 409–430.

    Article  Google Scholar 

  26. 26.

    S. K. Singh, U. Singh, and M. Kumar, Bayesian inference for exponentiated Pareto model with application to bladder cancer remission time, Transition in Statistics, 15 (2014), 403–426.

    Google Scholar 

  27. 27.

    A. A. Soliman, A. H. Abd-Ellah, N. A. Abou-Elheggag, and E. A. Ahmed, Modified Weibull model: A Bayes study using MCMC approach based on progressive censoring data, Reliability Engineering and System Safety, 100 (2012), 48–57.

    Article  Google Scholar 

  28. 28.

    H. R. Varian, A Bayesian approach to real estate assessment, North Holland, Amsterdam, (1975), 195–208.

    Google Scholar 

  29. 29.

    M. Xie, Y. Tang, and T. N. Goh, A modifiedWeibull extension with bathtub-shaped failure rate function, Reliability Engineering and System Safety, 76 (2002), 279–285.

    Article  Google Scholar 

  30. 30.

    T. Zhang and M. Xie, Failure data analysis with extended Weibull distribution, Communications in Statistics-Simulation and Computation, 36 (2007), 579–592.

    MathSciNet  Article  Google Scholar 

  31. 31.

    A. Zellner, Bayesian estimation and prediction using asymmetric loss functions, J. Am. Stat. Assoc., 81 (1986), 446–545.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgement

The authors would like to thank the Editor and reviewers for their valuable comments and suggestions to improve the presentation of the paper. The authors would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project No. 1440-142.

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Ziyad A. Alhussain or Essam A. Ahmed.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alhussain, Z.A., Ahmed, E.A. Estimation of exponentiated Nadarajah-Haghighi distribution under progressively type-II censored sample with application to bladder cancer data. Indian J Pure Appl Math 51, 631–657 (2020). https://doi.org/10.1007/s13226-020-0421-9

Download citation

Key words

  • Exponentiated Nadarajah-Haghighi distribution
  • maximum likelihood estimators
  • least squares estimators
  • Bayes estimators
  • bootstrap
  • Markov chain Monte Carlo

2010 Mathematics Subject Classification

  • 62F05
  • 62F10
  • 62F15
  • 62F40
  • 65C40