The increasing demand for flexible statistical models defined on the unit interval has led to the development of new distributions capable of capturing various tail behaviors and data shapes. In this study, we introduce an extended form of the power unit inverse Lindley distribution, offering greater modeling flexibility for bounded data. We explore its key statistical properties, including moments, skewness, kurtosis, quantile function, Fisher information matrix, extropy, and negative cumulative extropy. To assess parameter estimation, fifteen classical methods are implemented and evaluated through simulation, demonstrating high efficiency and low bias, even for small samples. The proposed model is then applied to two real datasets: annual inflation rates from 45 Asian countries and failure times of mechanical components. Comparative analysis with several existing unit distributions confirms the superior goodness-of-fit and practical applicability of the proposed model in both economic and engineering contexts.