On Robust and Non-Robust Modified Liu Estimation in Poisson Regression Model with Multicollinearity and Outliers

The Poisson regression model is widely used in statistical data modeling in diverse fields, such as epidemiology, economics, engineering, and sports. Its popularity stems from its ability to effectively describe the relationship between a statistical response variable and a set of explanatory variables. However, multicollinearity among predictors and outliers in the data can severely affect the reliability of parameter estimation, leading to inflated variances and biased results. Multicollinearity increases the sensitivity of estimators to small changes in the data, while outliers can disproportionately affect the model fit, especially in the maximum likelihood estimation framework. To address these issues, robust biased estimation techniques, such as the ridge estimator and modified ridge-type estimator, have been introduced to mitigate multicollinearity by introducing shrinkage parameters. However, the transformed M-estimator remains sensitive to outliers. In this study, we propose robust and non-robust modified Liu estimators for the Poisson regression model. The non-robust method reduces the effect of multicollinearity by developing a Liu estimator, while the proposed robust method combines transformed M-estimation with a modified form of the Liu estimator to simultaneously address multicollinearity and reduce the influence of outliers. We derive the theoretical properties of the estimators and investigate their performance through extended Monte Carlo simulations. The results indicate that the non-robust modified Liu estimator provides more stable and accurate estimates than maximum likelihood estimation, ridge estimator, and modified ridge-type estimators in the case of multicollinearity, but the robust version of the modified Liu estimator generally outperforms, especially in the presence of both multicollinearity and outliers. We also analyze a real-world dataset to demonstrate the practical value of the proposed methods.