Discrete Odd Weibull-Lindley Distribution Featuring Adaptable Hazard Rate and Dispersion Index

Theoretical Framework and Applications to SDG-Aligned Count Data

Abstract

This paper introduces the discrete odd Weibull--Lindley (DOWL) distribution, a flexible three-parameter model for analyzing complex count data frequently encountered in sustainable development research. The distribution is constructed by discretizing the continuous Lindley distribution within the discrete odd Weibull-G transformation framework. Its fundamental statistical and reliability properties are derived, including the identifiability of the mode and parameter effects, quantile function, hazard and reversed hazard rate functions, moments, information measures, residual and past lifetime functions, and equilibrium distribution. The DOWL distribution exhibits substantial flexibility, accommodating symmetric and asymmetric shapes, various kurtosis levels, and different dispersion patterns (over-, under-, and equi-dispersion). Its hazard rate function supports multiple aging behaviors, such as increasing, decreasing, bathtub, and unimodal shapes, making it suitable for diverse reliability and survival contexts. Parameter estimation is conducted via maximum likelihood, and extensive Monte Carlo simulations demonstrate satisfactory finite-sample performance of the estimators. The practical utility of the model is illustrated through three real data applications related to the United Nations Sustainable Development Goals: kidney cyst counts (SDG~3), European red mite infestation counts (SDGs~2 and~15), and daily COVID-19 mortality counts in Greece (SDG~3). Comparative analyses show that the DOWL distribution provides improved fit over competing discrete models, particularly for over-dispersed and highly skewed count data.