A Comparative Study of Estimation Methods for the Bell Regression Model under Multicollinearity

Abstract

The Bell regression model is a contemporary statistical framework for the analysis of count data, especially in instances of overdispersion, in contrast to the Poisson model and the Negative binomial regression model. However, the existence of multicollinearity among explanatory variables can cause instability in parameter estimations when employing the Maximum Likelihood Estimator (MLE), leading to diminished estimation accuracy.

This paper examines the issue of multicollinearity in the Bell regression model by evaluating various estimating techniques, including the Maximum Likelihood Estimator (MLE), Bell Ridge Regression (BRR), Bell Liu Regression (BLR), the K-L estimator, and the Jackknifed K-L estimator. The efficacy of these estimators was assessed utilizing the Mean Squared Error (MSE) criterion. A Monte Carlo simulation analysis was performed utilizing sample sizes (n = 50, 100, 150, 250), five explanatory variables (p = 5), and varying correlation levels (0.90, 0.95, 0.99). The simulation results indicated that the Bell Ridge regression estimator at the shrinkage parameter (k₁) achieved the lowest mean squared error (MSE) values in most instances relative to other estimators, indicating its superior performance in the presence of multicollinearity. Furthermore, real data from prostate cancer patients were analyzed. The results indicated that the Bell Ridge regression estimator outperformed the maximum likelihood estimator (MLE) and other estimators by attaining the lowest mean squared error (MSE), thereby demonstrating its capacity to deliver more stable and precise estimates, enhance the reliability of predicting disease-related outcomes, and effectively address issues of multicollinearity and overdispersion in the count data.