# A Flexible Bathtub-Shaped Failure Time Model: Properties and Associated Inference

## DOI:

https://doi.org/10.6092/issn.1973-2201/10025## Keywords:

Extended modified Weibull distribution, Maximum likelihood estimates, Bayesian estimates, Gibbs sampler, Tierney and Kadane's approximation, Highest posterior density intervals## Abstract

In this study, we introduce an extended version of the modified Weibull distribution with an additional shape parameter, in order to provide more flexibility to its density and the hazard rate function. The distribution is capable of modeling the bathtub-shaped, decreasing, increasing and the constant hazard rate function. The proposed model contains sub-models that are widely used in lifetime data analysis such as the modified Weibull, Chen, extreme value, Weibull, Rayleigh, and exponential distributions. We study its statistical properties which include the hazard rate function, moments and distribution of the order statistics. The parameters involved in the model are estimated by using maximum likelihood and the Bayesian method of estimation. In Bayesian estimation, we assume independent Gamma priors for the parameters and MCMC technique such as the Metropolis-Hastings algorithm within Gibbs sampler has been implemented to obtain the sample-based estimators and the highest posterior density intervals of the parameters. Tierney and Kadane (1986) approximation is also used to obtain Bayes estimators of the parameters. In order to highlight the relative importance of various estimates obtained, a simulation study is carried out. The usefulness of the proposed model is illustrated using two real datasets.

## References

M. V. AARSET (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36, no. 1, pp. 106–108.

C. ALEXANDER, G. M. CORDEIRO, E. M. ORTEGA, J. M. SARABIA (2012). Generalized beta-generated distributions. Computational Statistics & Data Analysis, 56, no. 6, pp. 1880–1897.

S. J. ALMALKI, J. YUAN (2013). A new modified Weibull distribution. Reliability Engineering & System Safety, 111, pp. 164–170.

M. ALMHEIDAT, F. FAMOYE, C. LEE (2015). Some generalized families of Weibull distribution: Properties and applications. International Journal of Statistics and Probability, 4, no. 3, p. 18.

M. AMINI, S. MIRMOSTAFAEE, J. AHMADI (2014). Log-gamma-generated families of distributions. Statistics, 48, no. 4, pp. 913–932.

L. J. BAIN (1974). Analysis for the linear failure-rate life-testing distribution. Technometrics, 16, no. 4, pp. 551–559.

M. BEBBINGTON, C.-D. LAI, R. ZITIKIS (2006). Useful periods for lifetime distributions with bathtub shaped hazard rate functions. IEEE Transactions on Reliability, 55, no. 2, pp. 245–251.

J. M. CARRASCO, E. M. ORTEGA, G. M. CORDEIRO (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics & Data Analysis, 53, no. 2, pp. 450–462.

M.-H. CHEN, Q.-M. SHAO (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8, no. 1, pp. 69–92.

Z. CHEN (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics & Probability Letters, 49, no. 2, pp. 155–161.

R. D. COOK, S. WEISBERG (2009). An Introduction to Regression Graphics, vol. 405. John Wiley & Sons, USA.

M. EL-MORSHEDY, F. S. ALSHAMMARI, A. TYAGI, I. ELBATAL, Y. S. HAMED, M. S. ELIWA (2021). Bayesian and frequentist inferences on a type I half-logistic odd Weibull generator with applications in engineering. Entropy, 23, no. 4, p. 446.

N. EUGENE, C. LEE, F. FAMOYE (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and methods, 31, no. 4, pp. 497–512.

S.GEMAN, D.GEMAN (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on pattern analysis and machine intelligence, 6, pp. 721–741.

W. K. HASTINGS (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, pp. 97–109.

A. JAMALIZADEH, A. ARABPOUR, N. BALAKRISHNAN (2011). A generalized skew two-piece skew-normal distribution. Statistical Papers, 52, no. 2, pp. 431–446.

C. LAI, M. XIE, D. MURTHY (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52, no. 1, pp. 33–37.

N. METROPOLIS, S. ULAM (1949). The Monte Carlo method. Journal of the American Statistical Association, 44, no. 247, pp. 335–341.

A. M. SARHAN, M. ZAINDIN (2009). Modified Weibull distribution. Applied Sciences, 11, pp. 123–136.

B. SINGH, R. GOEL (2018). Reliability estimation of modified Weibull distribution with type-II hybrid censored data. Iranian Journal of Science and Technology, Transactions A: Science, 42, no. 3, pp. 1395–1407.

B. SINGH, R. GOEL (2019). MCMC estimation of multi-component load-sharing system model and its application. Iranian Journal of Science and Technology, Transactions A: Science, 43, no. 2, pp. 567–577.

A. A. SOLIMAN, A. H. ABD-ELLAH, N. A. ABOU-ELHEGGAG, E. A. AHMED (2012). Modified Weibull model: A Bayes study using MCMC approach based on progressive censoring data. Reliability Engineering & System Safety, 100, pp. 48–57.

M. TAHIR, M. ZUBAIR, M. MANSOOR, G. M. CORDEIRO, M. AL˙IZADEHK, G. HAMEDANI (2016). A new Weibull-G family of distributions. Hacettepe Journal of Mathematics and Statistics, 45, no. 2, pp. 629–647.

L. TIERNEY, J. B. KADANE (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81, no. 393, pp. 82–86.

M. XIE, C. D. LAI (1996). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliability Engineering&System Safety, 52, no. 1, pp. 87–93.

M. XIE, Y. TANG, T. N. GOH (2002). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 76, no. 3, pp. 279–285.

K. ZOGRAFOS,N. BALAKRISHNAN (2009). On families of beta-and generalized gamma generated distributions and associated inference. Statistical Methodology, 6, no. 4, pp. 344–362.

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*81*(1), 65–92. https://doi.org/10.6092/issn.1973-2201/10025

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