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

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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.

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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

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Authors and Affiliations

  1. Department of Statistics, Central University of Rajasthan, Ajmer, Rajasthan, India

    Mahendra Saha

  2. Department of Statistics, Banaras Hindu University, Varanasi, India

    Abhimanyu Singh Yadav

Authors
  1. Mahendra Saha
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  2. Abhimanyu Singh Yadav
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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 10, 303–317 (2021). https://doi.org/10.1007/s41872-020-00162-9

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