Classical and Bayesian Inference for the Epanechnikov–Weibull Distribution Under Improved Adaptive Type-II Progressive Censoring, with Reliability and Extropy Analysis

The Epanechnikov–Weibull distribution (EpWD) is a flexible two-parameter lifetime model. Its hazard rate can be increasing or decreasing, and for some shape values below one, can also exhibit finite-time nonmonotonicity. This paper develops classical and Bayesian inference for the EpWD under the improved adaptive Type-II progressive censoring scheme (IAT-II PCS), a design that balances the objective of observing a prescribed number of failures with the requirement that the experiment terminate by a fixed upper time threshold. An alternative closed-form trigonometric representation of the quantile function is derived, providing a real-valued and convenient basis for random variate generation, and the scale-free weighted-extropy identity of the EpWD is exploited to obtain particularly simple inference for that measure. Maximum likelihood estimates are obtained together with an explicit observed information matrix, from which approximate confidence intervals for the parameters, reliability and hazard rate functions, and extropy and weighted extropy are constructed by the delta method. Bayesian estimates under gamma priors are computed by a Metropolis–Hastings algorithm with highest posterior density credible intervals; the extropy is summarised by its posterior median conditional on its existence region (shape above one-half), since its posterior mean is not finite near that pole. An extensive simulation study and two real engineering datasets evaluate the procedures. Bayesian estimators under a truth-centred informative prior show smaller errors for the shape parameter and the derived reliability measures in the simulated settings, particularly in small samples. The prior is an oracle benchmark rather than a practically available specification; among the implementable analyses, the maximum likelihood estimator remains competitive and is frequently preferable for the scale parameter. On the near-exponential electronic device data, the Akaike criterion favours the Gompertz and exponential models, while the EpWD provides the closest fit among the flexible two-parameter alternatives considered here.