This paper introduced and investigated the power unit inverse Lindley distribution (PUILD), a novel two-parameter generalization of the famous unit inverse Lindley distribution. Among its notable functional properties, the corresponding probability density function can be unimodal, decreasing, increasing, or right-skewed. In addition, the hazard rate function can be increasing, U-shaped, or N-shaped. The PUILD thus takes advantage of these characteristics to gain flexibility in the analysis of unit data compared to the former unit inverse Lindley distribution, among others. From a theoretical point of view, many key measures were determined under closed-form expressions, including mode, quantiles, median, Bowley's skewness, Moor's kurtosis, coefficient of variation, index of dispersion, moments of various types, and Lorenz and Bonferroni curves. Some important measures of uncertainty were also calculated, mainly through the incomplete gamma function. In the statistical part, the estimation of the parameters involved was studied using fifteen different methods, including the maximum likelihood method. The invariant property of this approach was then used to efficiently estimate different uncertainty measures. Some simulation results were presented to support this claim. The significance of the PUILD underlying model compared to several current statistical models, including the unit inverse Lindley, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, was illustrated by two applications using real datasets.

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.