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On Bayesian reliability estimation of a 1-out-of-k load sharing system model of modified Burr-III distribution

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Abstract

This article deals with classical and Bayesian estimation of the parameters of a 1-out-of-k load-sharing parallel system model in which each component’s lifetime follows modified Burr-III distribution. In the classical set up, the maximum likelihood estimates of the load-share parameters, system reliability and hazard rate functions with their bias, standard errors (SEs), average absolute bias (AABias) and width of the confidence interval along with coverage probability are obtained. Besides, two bootstrap confidence intervals for the parameters of the model are also obtained. Further, by assuming both gamma and non-informative priors for the unknown parameters, Bayes estimates of the parameters, system reliability and hazard rate functions with their bias, SEs, AABias and width of the Bayes credible interval along with coverage probability (\(C_{p}\)) are obtained. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. Finally, we analyze one real data set to illustrate the results derived.

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Correspondence to Azeem Ali.

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Appendix

Appendix

See Tables 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 and 26.

Table 5 ML estimates, width of ACIs and coverage probabilities for \(\alpha =2, \beta =1, \gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\)
Table 6 ML estimates, width of ACIs and coverage probabilities for \(\alpha =3, \beta =3, \gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\)
Table 7 Boot-p and boot-t estimates along with width of the confidence interval and coverage probabilities
Table 8 Boot-p and boot-t estimates along with width of the confidence interval and coverage probabilities
Table 9 Bayes estimates using gamma prior and non-informative prior
Table 10 Bayes estimates using gamma prior and non-informative prior
Table 11 Reliability estimates, width of ACIs and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using MLE
Table 12 Reliability estimates, width of ACIs and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using MLE
Table 13 Reliability estimates, width of ACIs and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using bootstrapping methods
Table 14 Reliability estimates, width of BCIs and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using bootstrapping methods
Table 15 Reliability estimates, width of HPD intervals and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using gamma priors
Table 16 Reliability estimates, width of HPD intervals and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using gamma priors
Table 17 Reliability estimates, width of HPD intervals and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using non-informative prior
Table 18 Reliability estimates, width of HPD intervals and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using non-informative prior
Table 19 Hazard rate estimates, width of ACIs and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using MLE
Table 20 Hazard rate estimates, width of ACIs and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using MLE
Table 21 Hazard rate estimates, width of BCIs and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using bootstrapping methods
Table 22 Hazard rate estimates, width of BCIs and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using bootstrapping methods
Table 23 Hazard rate estimates, width of HPD intervals and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using gamma priors
Table 24 Hazard rate estimates, width of HPD intervals and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using gamma priors
Table 25 Hazard rate estimates, width of HPD intervals and coverage probabilities for \(\alpha =2\), \(\beta =1\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using non-informative prior
Table 26 Hazard rate estimates, width of HPD intervals and coverage probabilities for \(\alpha =3\), \(\beta =3\), \(\gamma _{1} =0.2, \gamma _{2} =0.4, \gamma _{3} =0.6\) using non-informative prior

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Ali, A., Dey, S., Ur Rehman, H. et al. On Bayesian reliability estimation of a 1-out-of-k load sharing system model of modified Burr-III distribution. Int J Syst Assur Eng Manag 10, 1052–1081 (2019). https://doi.org/10.1007/s13198-019-00835-4

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