The Heavy-Tailed Inverse Power Lindley Type-I Model: Reliability Inference and Actuarial Applications

The analysis and modeling of asymmetric data present an interesting and important area of research across various applied sciences, particularly in lifetime studies, medical research, and financial analysis. In this work, we present the heavy-tailed inverse power Lindley Type-I (HTIPL-TI) distribution, a versatile three-parameter probability model. This distribution is derived by applying the generator of the Type-I heavy-tailed family to the inverse power Lindley model. The new model provides more flexibility in the shape of the inverse power Lindley with the addition of the shape parameter via the Type-I heavy-tailed-G family. The suggested distribution accommodates nonmonotonic patterns with a great versatility in capturing the features of lifetime data with increasing, U-shaped, N-shaped, upside-down bathtub-shaped, and reversed J-shaped. Some mathematical and statistical properties of the HTIPL-TI distribution were examined. We discuss the estimation of the distribution parameters and reliability functions (survival and hazard rate) for the HTIPL-TI by considering the maximum likelihood (ML) method along with their asymptotic confidence intervals. Bayesian estimators for the parameters and reliability functions (survival and hazard rate) are derived using gamma priors and both symmetric and asymmetric loss functions. Furthermore, the highest posterior credible intervals are created. Given the complex nature of various Bayesian estimates, the Markov Chain Monte Carlo method, which uses the Metropolis–Hastings algorithm, is employed. Monte Carlo simulation is used to evaluate the performance of the generated estimators, assessing accuracy by examining average interval length, coverage probability, and mean squared error. The results of the study confirmed that Bayes estimates are generally more appropriate than ML estimates. Also, the highest posterior density (HPD) credible intervals outperform the confidence intervals of the ML estimates based on average interval length and coverage probability in most situations. In most circumstances, the Bayesian estimates under the minimum expected loss function provide the best values corresponding to other loss functions. The flexibility and goodness-of-fit performance of the proposed model are also demonstrated by re-analyzing two actuarial real datasets and comparing its fit with those obtained by extended inverse Lindley, extended exponentiated inverse Lindley, inverse power Lindley, inverse Lindley, and alpha power-transformed inverse Lindley distributions.