Modelling the Proportions with Excessive Endpoints Based on a Generalized Lindley Binomial Model

This paper introduces the generalized Lindley binomial (GLB) distribution, a novel model for analyzing proportional data with excessive endpoint observations. The GLB distribution is derived by compounding the binomial distribution with a generalized three-parameter Lindley distribution, itself defined as a mixture of two gamma distributions with distinct rate parameters. We establish the probabilistic properties of the GLB distribution, including its probability mass function, factorial moments, mean, variance, moment generating function, and dispersion index, demonstrating its flexibility in modeling both under- and over-dispersed data as well as unimodal and bimodal shapes. Likelihood-based inference is developed for the GLB model, with and without covariates, using Fisher scoring and expectation-maximization (EM) algorithms. To improve estimation stability, a penalized EM algorithm incorporating Bayes-inspired penalties is proposed. Model diagnostics are addressed through Pearson and deviance residuals, as well as randomized quantile residual plots. Simulation studies are conducted to evaluate the performance of the estimation procedures under different scenarios. Finally, the practical utility of the GLB regression model is illustrated with the whitefly dataset, where it is shown to provide superior fit compared to existing endpoint-inflated binomial models.