Inference for a step-stress model under the inverted Z-Lindley distribution with progressive Type-Ⅱ censoring

We develop an explicit step–stress cumulative exposure model (CEM) formulation with closed–form equivalent times, analytic score functions, and analytic observed information for the one–parameter inverted Z–Lindley (IZL) lifetime distribution under progressive Type–Ⅱ censoring. Because the IZL quantile inverts in closed form via the Lambert $ W $ function, the equivalent start time linking the two stress stages is itself closed form, giving an explicit rather than implicit likelihood, from which we derive the exact score, prove strict concavity of the constant–stress log–likelihood for complete samples, and obtain the analytic observed information matrix, validated against finite differences. The construction extends to a multiple–step generalization; numerical change–time optimization under $ c $–, $ D $–, and $ A $–optimality for a representative complete–sample design; and a link–function extrapolation to an untested design stress with delta–method variance. Inference is triangulated across maximum likelihood, Wald, percentile bootstrap, and gamma–prior Bayesian analysis, cross–checked by the Tierney–Kadane approximation. A Monte Carlo study shows finite-sample bias and mean squared error (MSE) decreasing with sample size and near-nominal coverage for the Wald, Bayesian, and bootstrap intervals in most scenarios as well as accurate design–stress extrapolation under a correctly specified link, which degrades sharply once the link is misspecified. The principal estimation and uncertainty–quantification procedures are illustrated using a real, Type–Ⅰ–censored step–stress data set, and design–stress extrapolation is assessed separately through a four–level simulation; the real–data example thus validates the broader CEM/IZL modeling and inference approach rather than the progressive Type–Ⅱ censoring likelihood itself. On a small unmanned aerial vehicle (SUAV) reliability data set, a bootstrap–calibrated goodness–of–fit test finds no evidence against the fitted model, and IZL is among a small cluster of equally well–supported models by AICc – supporting IZL as a credible, closed-form candidate for step–stress reliability analysis.