A framework for modeling unit-interval data based on mixtures of Lindley distributions with engineering applications

Data supported on the unit interval naturally arise in the analysis of proportions, rates, indices, and bounded engineering measures. Such data often exhibit asymmetry, boundary concentration, and latent heterogeneity. We propose a unified two-component mixture framework based on the unit-Lindley (UL) and new unit-Lindley (NUL) distributions. Each component may belong to either family, leading to four possible configurations: UL+UL, UL+NUL, NUL+UL, and NUL+NUL. The proposed class is parameterized in terms of component means, admits flexible density shapes, and is closed under the reflection transformation \(Y \mapsto 1-Y\) , which provides structural symmetry. We derive the density, moment structure, identifiability conditions, and likelihood-based inference. Parameter estimation is addressed through a deterministic expectation-maximization algorithm with closed-form maximization-step updates, together with a stochastic variant designed to improve exploration of the likelihood surface. Standard errors and asymptotic confidence intervals are obtained from the observed information matrix using the missing-information principle, complemented by parametric bootstrap intervals. A Monte Carlo study evaluates bias, mean squared error, coverage probability, and confidence interval length under different sample sizes and mixing proportions. Three real-data applications show that the proposed mixtures provide competitive, and in some cases superior, fits compared with standard unit-interval models, including the recently proposed unit one-parameter polynomial exponential distribution. A Python implementation is also provided to facilitate reproducible use of the methodology.