Linearized Ridge-Type Estimator for the Poisson Modification of the Quasi-Lindley Regression Model

Abstract

Abstract This paper develops a linearized ridge-type estimator for the Poisson-modification of the quasi-Lindley regression model (PMQL-RM), which is designed for overdispersed count responses frequently encountered in epidemiology, insurance, and related fields. The PMQL-RM extends classical Poisson regression by incorporating a flexible mixed Poisson distribution with two dispersion parameters, thereby accommodating substantial overdispersion while retaining a generalized linear model structure and maximum likelihood estimation via iteratively weighted least squares. However, in many practical applications, the explanatory variables are highly correlated, leading to multicollinearity that inflates the variance of the PMQL maximum likelihood estimator (PMQL-MLE) and yields unstable inference. Existing shrinkage estimators for this model, including the PMQL ridge estimator (PMQL-RRE), the PMQL Liu estimator (PMQL-LE), and the PMQL Liu-type estimator (PMQL-LTE), partially address this issue by introducing biasing parameters, but there remains room for further reduction in mean squared error (MSE). Motivated by the superior MSE properties of the linearized ridge regression estimator in linear and generalized linear models, we propose its extension to the PMQL-RM, termed the PMQL linearized ridge estimator (PMQL-LRE). We derive the bias, variance–covariance matrix, and matrix MSE of the PMQL-LRE and establish theoretical conditions under which it dominates PMQL-MLE, PMQL-RRE, PMQL-LE, and PMQL-LTE in both matrix and scalar MSE senses. A Monte Carlo simulation study, conducted under varying levels of multicollinearity, sample sizes, numbers of predictors, and dispersion parameter settings, demonstrates that the PMQL-LRE generally achieves the smallest MSE, especially under severe multicollinearity. An empirical application to real overdispersed count data further illustrates the practical advantages of the proposed estimator in terms of improved estimation accuracy and more stable coefficient estimates.