Eslam Abdelhakim Seyam
Department of Insurance and Risk Management, College of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
E. M. Moustafa
Department of Statistics and Insurance, Faculty of Commerce, Zagazig University, Zagazig
Despite the growing interest in one-parameter lifetime distributions from the Lindley family, no study has addressed parameter estimation and reliability inference for the Ishita distribution under Type-II hybrid censoring. This paper develops a complete inferential framework for the Ishita distribution based on Type-II hybrid censored data, encompassing maximum likelihood estimation, Bayesian analysis via Metropolis-Hastings MCMC, and parametric bootstrap methods. The MLE is obtained through Newton-Raphson optimization using the maxLik package, with asymptotic confidence intervals derived from the observed Fisher information under both normal approximation and log-transformation. Bayesian estimation employs a log-normal random walk Metropolis-Hastings sampler under two gamma priors—a non-informative Gamma(0.001, 0.001) and an informative Gamma(2, 2)—yielding posterior summaries with equal-tailed Bayesian credible intervals and highest posterior density intervals for the parameter θ, the reliability function R(x), and the hazard function h(x). A Monte Carlo simulation study with M = 200 replications across six censoring configurations demonstrates that the RMSE of the estimator of θ decreases from 0.1346 (n = 20) to 0.0929 (n = 50), a 31% reduction, while the informative Prior 2 consistently yields the lowest RMSE across all quantities. Coverage probabilities range from 0.910 to 0.975 for ACI-NA and from 0.915 to 0.975 for HPD intervals, with HPD intervals achieving the shortest average lengths. Two real data applications are presented: carbon fiber tensile strength data (n = 50, d = 40) and airplane component failure data (n = 36, d = 30). The MLE and Bayesian posterior means agree to within 0.52%, confirming estimation consistency. Goodness-of-fit testing via Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises statistics indicates that the Ishita distribution does not provide an adequate fit for either dataset (p < 0.001), suggesting the need for generalized extensions. Model comparison is conducted using AIC, BIC, and DIC criteria.
