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Open Access Research Article

Conjugate and objective priors for the parameter and cumulative function under bathtub and increased shapes of hazard function

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pp. 235–255Vol. 28Issue 2March 2025DOI: 10.47974/JSMS-969XML
Received:
14 Sep 2021
Accepted:
05 Apr 2022
Published Online:
01 Mar 2025
Article type:
Research Article
Language:
EN
Article no.:
JSMS-969
Pages:
235–255

Abstract

We present a full Bayesian inference for the parameter and the associated reliability function of the Exponentiated Gamma distribution. This distribution has the advantage of being uni-parametric and its hazard function takes on increasing and bathtub shapes. We show that dealing with Jeffreys, uniform and gamma priors for the parameter then the corresponding posterior distributions result in a Gamma. Besides, the posterior distributions of cumulative function for a fixed lifetime are Negative Log-Gamma (NLG) distributions under transformation of parameter and also when Jeffreys, uniform and NGL priors are assigned to the cumulative function. This way, we show that the families of gamma and NLG distributions provide conjugate prior distribution for the parameters of interest. We also construct credible and HPD intervals in a simple closed-form. We further derive the predictive distribution and interval of future observation assuming gamma posterior distribution for the parameter.

Keywords

Subject Classifications

62F15

References

[1] A. Berhane, "Discretization of some continuous distribution and Their Applications," Eritrea: Master thesis submitted to Department of Statistics, College of Science, Eritrea Institute of Technology (2018).
[2] Z. Chen, "A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function," Statistics and Probability Letters, vol. 49, issue 2, pp. 155-161 (2000).
[3] N. Feroze, M. Aslam, "Bayesian analysis of exponentiated gamma distribution under type II censored samples," Scientic Journal of Pure and Applied Sciences, vol. 1, issue 1, pp. 30-39 (2012).
[4] R. C. Gupta, R. D. Gupta, P. L. Gupta, "Modeling failure time data by Lehman alternatives," Commun. Statist.-Theory, Meth. vol. 27, issue 4, pp.887-904 (1998).
[5] H. Jereys, "Theory of probability," 3rd rev. ed., London: Oxford University Press (1967).
[6] R. U. Klan, D. Kumar, "Lower generalized order statistics from exponentiated Gamma distribution and its characterization", ProbStat Forum, vol. 4, pp. 1224 (2011).
[7] H. Martz and R. Waller, "Bayesian Reliability Analysis," Wiley, New York (1982).
[8] A.I. Shawky, R.A. Bakoban, "Bayesian and Non-Bayesian Estimations on the Exponentiated Gamma Distribution", Applied Mathematical Sciences, vol. 2, pp. 2521-2530 (2008).
[9] A.I. Shawky, R.A. Bakoban, "Order statistics from exponentiated gamma distribution and associated inference," The International Journal of Contemporary Mathematical Sciences, vol. 4, issue 2, pp. 71-91 (2009(a)).
[10] A.I. Shawky, R.A. Bakoban, "Conditional expectation of certain distributions of record values," Int. Journal of Math. Analysis, vol. 3, issue 17, pp. 829-838 (2009(b)).
[11] A.I. Shawky, R.A. Bakoban, "Certain characterizations of the exponentiated gamma distribution. Journal of Statistics Sciences. Vol 3, issue 2, pp.151-164 (2011).
[12] R. Shehla, A. A. Khan, "Reliability analysis using an exponential power model with bathtub-shaped failure rate function: a Bayes study," Springer Plus, vol. 5, n. 1076, pp. 1-22 (2016).
[13] U. Singh, S. K. Singh, A. S. Yadav, "Bayesian Estimation for Exponentiated Gamma Distribution Under Progressive Type-II Censoring Using Different Approximation Techniques," Journal of Data Science, vol 13, pp. 549-566 (2015).
[14] R. M. Smith, L. J. Bain, "An exponential power life-test distribution," Communications in Statistics, vol. 4, pp. 469-481 (1975).

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