The Inverted Hjorth Distribution and Its Applications in Environmental and Pharmaceutical Sciences

This study introduces an inverted version of the three-parameter Hjorth lifespan model, characterized by one scale parameter and two shape parameters, referred to as the inverted Hjorth (IH) distribution. This asymmetric distribution can fit various positively skewed datasets more accurately than several existing models in the literature, as it can accommodate data exhibiting an inverted (upside-down) bathtub-shaped hazard rate. We derive key properties of the model, including quantiles, moments, reliability measures, stress–strength reliability, and order statistics. Point estimation of the IH model parameters is performed using maximum likelihood and Bayesian approaches. Moreover, for interval estimation, two types of asymptotic confidence intervals and two types of Bayesian credible intervals are obtained using the same estimation methodologies. As an extension to a complete sampling plan, Type-II censoring is employed to examine the impact of data incompleteness on IH parameter estimation. Monte Carlo simulation results indicate that Bayesian point and credible estimates outperform those obtained via classical estimation methods across several precision metrics, including mean squared error, average absolute bias, average interval length, and coverage probability. To further assess its performance, two real datasets are analyzed: one from the environmental domain (minimum monthly water flows of the Piracicaba River) and another from the pharmacological domain (plasma indomethacin concentrations). The superiority and flexibility of the inverted Hjorth model are evaluated and compared with several competing models. The results confirm that the IH distribution provides a better fit than several existing lifetime models—such as the inverted Gompertz, inverted log-logistic, inverted Lomax, and inverted Nadarajah–Haghighi distributions—making it a valuable tool for reliability and survival data analysis.