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


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.

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Akaike information criterion


Average estimate


Average width


Bayesian information criterion

\(\mathcal {BCPB}\) ::

Bias-corrected percentile bootstrap


Bootstrap confidence interval


Consistent AIC


Confidence interval


Coverage probability


Cumulative distribution function


Entropy loss function


Exponential distribution


Highest posterior density


Linex loss function


Least squares estimator


Markov Chain Monte Carlo


Maximum product of spacings estimator


Mean time to system failure


Maximum likelihood estimator


Mean squared error


Non-linear minimization


Ordinary LSE

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

Percentile bootstrap


Probability density function


Precautionary loss function


Reliability function


Standard error

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

Standard bootstrap


Squared error loss function


Xgamma distribution


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Correspondence to Abhimanyu Singh Yadav.

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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).

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

Mathematics Subject Classification

  • 62E05
  • 62B10
  • 62F15