Bayesian Analysis of the Discrete Two-Parameter Bathtub Hazard Distribution

Year 2019, , 183 - 192, 26.12.2019

Ammar Sarhan

https://doi.org/10.33187/jmsm.570880

Cited By: 1

Abstract

A new discrete two-parameter bathtub hazard distribution is proposed by Sarhan \cite{Sarhan-2017}. This paper uses Bayes method to estimate the two unknown parameters and the reliability measures of this distribution. The joint posterior distribution of the model parameters cannot be obtained in a convenient form. Therefore, numerical techniques are  needed. We apply four Bayesian numerical methods to get random draws from the joint posterior distribution to be used to estimate the model parameters and its reliability measures without deriving the actual joint posterior distribution.  It is assumed here that the two model parameters are priori independent random variables with beta and gamma distributions. Two scenarios for the hyperparameters are applied to compare their contributions on the Bayesian inferences. Two real data sets are re-analyzed using the Bayesian techniques applied here. A simulation study is performed to investigate the properties of the methods applied.

Keywords

Data analysis, Maximum likelihood estimator, Probability models, Reliability measures

References

• Aarset MV. How to identify a bathtub hazard rate. IEEE Trans. Reliability 1987; R-36; 106-108.
• Almalki SJ, Nadarajah S. A new discrete modified Weibull distribution, IEEE Transactions on Reliability 2014; 63(1); 68-80.
• Bebbington M, Lai C-D, Wellington M, Zitikis R. The discrete additive Weibull distribution: A bathtub-shaped hazard fordiscontinuous failure data, Reliability Engineering and System Safety 2012; 106; 37-44. Gelman, A, Carlin J, Stern H, Rubin D. Bayesian Data Analysis. New York: Chapman and Hall; 2003.
• Givens G, Hoeting J. Computational Statistics, Hoboken. NJ: John Wiley and Sons; 2005.
• Gnedenko B, Ushakov I. Probabilistic Reliability Engineering. John Wiley \& Sons, New York; 1995.
• Kapur KC, Lamberson LR. Reliability and in Engineering Design. John Wiley \& Sons, New York; 1976.
• Lawless JF. Statistical Models and Methods for Lifetime Data. John Wiley \& Sons, New York; 1982.
• Monahan J. Numerical Methods of Statistics. Cambridge: Cambridge University Press; 2001.
• Nakagawa T, Osaki S. The Discrete Weibull Distribution. IEEE Transactions on Reliability 1975; R-24(5); 300-301.
• Nooghabi MS, Roknabady AR, Borzadaran GM. Discrete modified Weibull distribution. Metron 2011; LXIX; 207-222.
• Robert C, Casella G. Monte Carlo Statistical Methods. New York: Springer; 2014.
• Sarhan AM. A two-parameter discrete distribution with a bathtub hazard shape. Communications for Statistical Applications and Methods 2017; Vol. 24(1); 15-27.
• Sinha SK. Reliability and Life testing. Wiley Eastern Ltd, New Delhi; 1986.
• Wang FK. A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliability Engineering and System Safety 2000; 70; 305-312.
• Xie M, Lai CD. Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliability Engineering System Safety 1995; 52: 87-93.