A new Liu-type estimator in a mixed Poisson regression model.

Mixed Poisson regression models (MPRMs) are widely used for analyzing overdispersed count data. However, the presence of multicollinearity among explanatory variables poses challenges when estimating regression coefficients using the maximum likelihood estimator (MLE), leading to inflated variances. The Poisson Modification of the Quasi-Lindley regression model (PMQLRM), a recently introduced alternative within MPRMs, faces similar issues. To address this, we propose a Liu-type estimator for the PMQLRM as an effective remedy for multicollinearity. Several existing methods are utilized to estimate the Liu-type parameter, and the theoretical superiority conditions of the proposed estimator over the MLE, ridge regression estimator, and Liu estimator are established using the scalar mean squared error (MSE) criterion. A Monte Carlo simulation study is conducted to compare the performance of different estimators based on the MSE. Additionally, a real-world dataset is analyzed to demonstrate the practical advantages of the proposed method. The findings indicate that the Poisson-modification of the Quasi-Lindley Liu-type estimator outperforms the MLE and other biased estimators when multicollinearity is present, offering a more stable and reliable alternative for parameter estimation in mixed Poisson regression models.