Estimation of the reliability characteristics by using classical and Bayesian methods of estimation for xgamma distribution

Abstract

In this article, estimation of the reliability characteristics viz., mean time to system failure \(M(t; \alpha )\), reliability function \(R(t; \alpha )\) is considered for xgamma lifetime distribution. First, four different methods of estimation of the reliability characteristics for specified value of time t are addressed from frequentist approaches and compared them in terms of their respective mean squared errors using extensive numerical simulations. Second, we compared three bootstrap confidence intervals (BCIs) including standard bootstrap, percentile bootstrap and bias-corrected percentile bootstrap. Third, Bayesian estimation is considered under three loss functions using gamma prior for the considered model. Fourth, we obtained highest posterior density (HPD) credible intervals of \(M(t; \alpha )\) and \(R(t; \alpha )\). Monte Carlo simulation study has been carried out to compare the performances of the classical BCIs and HPD credible intervals of \(M(t; \alpha )\) and \(R(t; \alpha )\) in terms of average widths and coverage probabilities. Finally, a real data set has been analyzed for illustrative purpose.

This is a preview of subscription content, access via your institution.

Abbreviations

AIC:

Akaike information criterion

AE:

Average estimate

AW:

Average width

BIC:

Bayesian information criterion

\(\mathcal {BCPB}\) ::

Bias-corrected percentile bootstrap

BCI:

Bootstrap confidence interval

CAIC:

Consistent AIC

CI:

Confidence interval

CP:

Coverage probability

CDF:

Cumulative distribution function

ELF:

Entropy loss function

ED:

Exponential distribution

HPD:

Highest posterior density

LLF:

Linex loss function

LSE:

Least squares estimator

MCMC:

Markov Chain Monte Carlo

MPSE:

Maximum product of spacings estimator

MTSF:

Mean time to system failure

MLE:

Maximum likelihood estimator

MSE:

Mean squared error

NLM:

Non-linear minimization

OLSE:

Ordinary LSE

\({{\mathcal {P}}}{{\mathcal {B}}}\) :

Percentile bootstrap

PDF:

Probability density function

PLF:

Precautionary loss function

RF:

Reliability function

SE:

Standard error

\({{\mathcal {S}}}{\mathcal {B}}\) :

Standard bootstrap

SELF:

Squared error loss function

XGD:

Xgamma distribution

References

  1. Calabria R, Pulcini G (1996) Point estimation under asymmetric loss functions for left-truncated exponential samples. Commun Stat Theory Method 25(3):585–600

    MathSciNet  Article  Google Scholar 

  2. Cheng RCH, Amin NAK (1979) Maximum product-of-spacings estimation with applications to the log-normal distribution. University of Wales IST, math report, pp 79–1

  3. Cheng RCH, Amin NAK (1983) Estimating parameters in continuous univariate distributions with a shifted origin. J R Stat Soc Ser B Stat Methodol 3:394–403

    MathSciNet  Article  MATH  Google Scholar 

  4. Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 6:69–92

    MathSciNet  Google Scholar 

  5. Dey S, Dey T, Kundu D (2014) Two-parameter Rayleigh distribution: different methods of estimation. Am J Math Manag Sci 33:55–74

    Google Scholar 

  6. Dey S, Ali S, Park C (2015) Weighted exponential distribution: properties and different methods of estimation. J Stat Comput Simul 85:3641–3661

    MathSciNet  Article  Google Scholar 

  7. Dennis JE, Schnabel RB (1983) Numerical methods for unconstrained optimization and non-linear equations. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  8. Efron B (1982) The Jackknife, the bootstrap and other re-sampling plans. In: CBMS-NSF Regional Conference Series in Applied Mathematics. Stanford University, Stanford, California

  9. Gross AJ, Clark VA (1975) Survival distributions: reliability applications in the biomedical sciences. Wiley, New York

    Google Scholar 

  10. Mann NR (1968) Point and interval estimation procedures for the two-parameter Weibull and extreme value distributions. Technometrics 10:231

    MathSciNet  Article  Google Scholar 

  11. Maiti K, Kayal S (2019) Estimation of parameters and reliability characteristics for a generalized Rayleigh distribution under progressive type-II censored sample. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2019.1630431

    Article  Google Scholar 

  12. Thoman DR, Bain LJ, Antle CE (1970) Maximum likelihood estimation, exact confidence intervals for reliability and tolerance limits in the Weibull distribution. Technometrics 12:363–72

    Article  Google Scholar 

  13. Lawless JF (1972) Confidence interval estimation for the parameters of the Weibull distribution. Utilitas Math. 2:71–87

    MathSciNet  MATH  Google Scholar 

  14. Lawless JF (1978) Confidence interval estimation for the Weibull and extreme value distributions. Technometrics 20(4):355–364

    MathSciNet  Article  Google Scholar 

  15. Roberts G0, Smith AEM (1994) Simple conditions for the convergence of the Gibbs sampler and Metropolis Hastings algorithms. Stoch Process Applic 49:207–216

    MathSciNet  Article  Google Scholar 

  16. Sen S, Maiti SS, Chandra N (2016) The xgamma distribution: statistical properties and application. J Mod Appl Stat Methods 15(1):774–788

    Article  Google Scholar 

  17. Saha M, Dey S, Yadav AS, Kumar S (2019a) Classical and Bayesian inference of \(C_{py}\) for generalized Lindley distributed quality characteristic. Qual Reliab Eng Int. https://doi.org/10.1002/qre.2544

    Article  Google Scholar 

  18. Sen S, Chandra N, Maiti SS (2018) Survival estimation in Xgamma distribution under progressively type-II right censored scheme. Model Assist Stat Appl 13(2):107–121. https://doi.org/10.3233/MAS-180423

    Article  Google Scholar 

  19. Surles JG, Padgett WJ (2001) Inference for reliability and stress-strength for a scaled Burr type X distribution. Lifetime Data Anal 7(2):187–200. https://doi.org/10.1023/A:1011352923990

    MathSciNet  Article  MATH  Google Scholar 

  20. Singh SK, Singh U, Yadav AS (2014) Bayesian estimation of Marshall–Olkin extended exponential parameters under various approximation techniques. Hecettepe J Math Stat 43(2):1–13

    MathSciNet  Article  Google Scholar 

  21. Singh SK, Singh U, Yadav AS (2015) Reliability estimation and prediction for extension of exponential distribution using informative and non-informative priors. Int J Syst Assur Eng Manag 6(4):466–478

    Article  Google Scholar 

  22. Singh SK, Singh U, Yadav AS (2016) Reliability estimation for inverse Lomax distribution under type-II censored data using Markov chain Monte Carlo method. Int J Math Stat 17(1):128–146

    MathSciNet  MATH  Google Scholar 

  23. Upadhyay SK, Vasishta N, Smith AFM (2001) Bayes inference in life testing and reliability via Markov chain Monte Carlo simulation. Sankhya Indian J Stat Ser A 63(1):15–40

    MATH  Google Scholar 

  24. Yadav AS, Saha M, Singh SK, Singh U (2018) Bayesian estimation of the parameter and the reliability characteristics of xgamma distribution using type-II hybrid censored. Life Cycle Reliab Saf Eng. https://doi.org/10.1007/s41872-018-0065-5

    Article  Google Scholar 

  25. Yadav AS, Bakouch HS, Chesneau C (2019) Bayesian estimation of the survival characteristics for Hjorth distribution under progressive type-II censoring. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2019.1659363

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Abhimanyu Singh Yadav.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Saha, M., Yadav, A.S. Estimation of the reliability characteristics by using classical and Bayesian methods of estimation for xgamma distribution. Life Cycle Reliab Saf Eng (2021). https://doi.org/10.1007/s41872-020-00162-9

Download citation

Keywords

  • Classical and Bayesian inference
  • Bootstrap confidence interval
  • Bayes credible interval

Mathematics Subject Classification

  • 62E05
  • 62B10
  • 62F15