This study introduces and investigates the truncated unit exponentiated Ailamujia (TUEA) distribution within the framework of a progressive Type–Ⅱ censoring scheme. By incorporating truncation on the unit interval, the proposed model extends the unit exponentiated Ailamujia distribution, significantly enhancing its flexibility for modeling bounded lifetime data. We derive the fundamental mathematical properties of the TUEA model, including the probability density function, the cumulative distribution function, reliability measures, and hazard rate functions. Statistical inference for the model parameters is developed within both frequentist and Bayesian frameworks using progressive Type–Ⅱ censored data. The maximum likelihood estimates are computed through the Newton–Raphson iterative algorithm, whereas Bayesian inference is carried out under symmetric squared error and asymmetric LINEX loss functions. Point estimates and highest posterior density credible intervals are obtained via Markov chain Monte Carlo (MCMC) sampling procedures. In addition, a comprehensive Monte Carlo simulation study is conducted to investigate the finite sample performance of the new estimators in terms of bias, mean square error, and confidence interval coverage probability. In addition, the practical usefulness of the TUEA distribution is shown via an analysis of a real-life data sets, where the new model exhibits a better fit than competing models.

The log-Lindley exponential (LLE), a bounded variant of the Lindley exponential distribution, is introduced using the inverse exponential transformation method. The LLE demonstrates significant flexibility, generating novel and well-known distributions with various parameters and providing support, including log-Lindley, Lindley, and two-parameter Lindley distributions, and exhibiting diverse density shapes, including right-skewed, approximately symmetric, left-skewed, decreasing, U-shaped, and increasing, while the hazard rate can be increasing, J-shaped, and bathtub-shaped. We explore several important statistical properties of the LLE model, including moments, entropy, quantile function, actuarial measures, mean residual life function, and stochastic orderings, which enhance its applicability in practical settings. Our study presents nine distinct frequentist strategies for estimating the model's parameters. In addition, the performance of the proposed estimation method is studied and evaluated through extensive numerical simulations. Finally, applications to real-world datasets from industrial engineering and epidemiology reveal the LLE model's practical utility and superiority, with it outperforming other existing models in terms of goodness-of-fit and flexibility.