UNIFIED HYBRID CENSORING FOR THE XGAMMA DISTRIBUTION: FREQUENTIST, BOOTSTRAP, AND BAYESIAN ESTIMATION WITH INDUSTRIAL APPLICATIONS
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2026
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| author | Eslam Abdelhakim Seyam and E. M. Moustafa |
| author_facet | Eslam Abdelhakim Seyam and E. M. Moustafa |
| contents | <p class="MsoNormal"><strong><em><span>In spite of the growing interest in the Xgamma distribution as a versatile one-parameter lifetime distribution, the inferential properties of the Xgamma distribution under unified hybrid censoring (UHC) have not been studied. In this paper, we develop both classical and Bayesian inference procedures for the Xgamma distribution under unified hybrid censoring. We compute the maximum likelihood estimates using the BFGS optimization algorithm after a logarithmic transformation of the likelihood function. We also develop asymptotic normal, asymptotic normal-log, percentile bootstrap, and bootstrap-t confidence interval methods for interval estimation. In addition, we develop a random walk Metropolis–Hasting’s algorithm using a logarithmic transformation and Robbins–Monro adaptation for the Bayesian inference under two gamma priors: a noninformative "Gamma"(0.001,0.001) and a data-adaptive "Gamma"(1,1/θ </span></em></strong><strong><em><span>̂</span></em></strong><strong><em><span>_"MLE”). In the simulation study, we have conducted nine plans under three groups of design with 200 replicates each. We have demonstrated that the RMSE of the estimator θ </span></em></strong><strong><em><span>̂</span></em></strong><strong><em><span> reduces from 0.1181 to 0.1048 when the sample size increases from n=20 to n=50, which is a reduction of 11.3%. In addition, the Bayesian estimator under Prior 2 has the least RMSE in all the plans. We have also demonstrated that the coverage probabilities of the ACI-NL interval lie in the range (0.510, 0.750) and perform better than other methods of interval estimation. We have also provided two examples using the failure data of Kevlar 49/epoxy strand and petroleum refinery equipment failure data. In these examples, the MLE and the Bayesian estimates of the parameters under the gamma prior lie within a relative error of 0.35%. We have also conducted the Kolmogorov–Smirnov test to check the goodness of fit of the Xgamma distribution to the data and have obtained p=0.0035 and p=0.0113 for the first and second datasets, respectively. We have used the criteria of AIC, BIC, and DIC to select the best model.</span></em></strong></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19746296 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | UNIFIED HYBRID CENSORING FOR THE XGAMMA DISTRIBUTION: FREQUENTIST, BOOTSTRAP, AND BAYESIAN ESTIMATION WITH INDUSTRIAL APPLICATIONS Eslam Abdelhakim Seyam and E. M. Moustafa Xgamma Distribution, Unified Hybrid Censoring, Maximum Likelihood Estimation, Bayesian Inference, MCMC, Bootstrap Confidence Intervals, Reliability Function, Hazard Rate. <p class="MsoNormal"><strong><em><span>In spite of the growing interest in the Xgamma distribution as a versatile one-parameter lifetime distribution, the inferential properties of the Xgamma distribution under unified hybrid censoring (UHC) have not been studied. In this paper, we develop both classical and Bayesian inference procedures for the Xgamma distribution under unified hybrid censoring. We compute the maximum likelihood estimates using the BFGS optimization algorithm after a logarithmic transformation of the likelihood function. We also develop asymptotic normal, asymptotic normal-log, percentile bootstrap, and bootstrap-t confidence interval methods for interval estimation. In addition, we develop a random walk Metropolis–Hasting’s algorithm using a logarithmic transformation and Robbins–Monro adaptation for the Bayesian inference under two gamma priors: a noninformative "Gamma"(0.001,0.001) and a data-adaptive "Gamma"(1,1/θ </span></em></strong><strong><em><span>̂</span></em></strong><strong><em><span>_"MLE”). In the simulation study, we have conducted nine plans under three groups of design with 200 replicates each. We have demonstrated that the RMSE of the estimator θ </span></em></strong><strong><em><span>̂</span></em></strong><strong><em><span> reduces from 0.1181 to 0.1048 when the sample size increases from n=20 to n=50, which is a reduction of 11.3%. In addition, the Bayesian estimator under Prior 2 has the least RMSE in all the plans. We have also demonstrated that the coverage probabilities of the ACI-NL interval lie in the range (0.510, 0.750) and perform better than other methods of interval estimation. We have also provided two examples using the failure data of Kevlar 49/epoxy strand and petroleum refinery equipment failure data. In these examples, the MLE and the Bayesian estimates of the parameters under the gamma prior lie within a relative error of 0.35%. We have also conducted the Kolmogorov–Smirnov test to check the goodness of fit of the Xgamma distribution to the data and have obtained p=0.0035 and p=0.0113 for the first and second datasets, respectively. We have used the criteria of AIC, BIC, and DIC to select the best model.</span></em></strong></p> |
| title | UNIFIED HYBRID CENSORING FOR THE XGAMMA DISTRIBUTION: FREQUENTIST, BOOTSTRAP, AND BAYESIAN ESTIMATION WITH INDUSTRIAL APPLICATIONS |
| topic | Xgamma Distribution, Unified Hybrid Censoring, Maximum Likelihood Estimation, Bayesian Inference, MCMC, Bootstrap Confidence Intervals, Reliability Function, Hazard Rate. |
| url | https://doi.org/10.5281/zenodo.19746296 |