## 6 Conclusions  
The PIG regression model is widely used for modeling count data with overdispersion. The model regression estimates are mostly estimated using the PIGMLE. However, multicollinearity is a threat to the performance of PIGMLE. To address this issue, we proposed a novel biased estimator: the modified Kibria-Lukman estimator (\(\hat{\mathbb{a}}^{PIG-MKLE}\)). Through theoretical analysis and extensive simulations, we evaluate their performance. The results demonstrate that the proposed estimator exhibits superior efficiency, as evidenced by a reduced MSE compared to traditional approaches. These findings suggest that our proposed estimators provide a more robust alternative for PIG regression in the presence of multicollinearity.  
An example study was conducted, examining squirrel behavior and forest characteristics across different plots in Abernethy Forest, Scotland. The proposed estimator has the lowest PMSE and PMAE among the shrinkage estimators. The findings further support the superiority of the proposed method in handling multicollinearity while maintaining predictive accuracy. Our findings enhance the existing research on count regression models and provide valuable analytical tools for researchers across various disciplines seeking robust estimation methods. In future studies, we aim to enhance the effectiveness of these estimators by incorporating techniques such as partial least squares and principal component regression.  
Conceptualization; All authors; methodology, All authors; formal analysis, All authors; writing--original draft preparation, All authors.; writing--review and editing, All authors; visualization, A.F.L.; supervision, All authors. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability Statement: Data will be made available upon request.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Coef.** & \(\hat{\mathbb{a}}^{\text{PIG}-\text{MLE}}\) & \(\hat{\mathbb{a}}^{\text{PIG}-\text{RRE}}\) & \(\hat{\mathbb{a}}^{\text{PIG}-\text{LE}}\) & \(\hat{\mathbb{a}}^{\text{PIG}-\text{ISE}}\) & \(\hat{\mathbb{a}}^{\text{PIG}-\text{TPE}}\) & \(\hat{\mathbb{a}}^{\text{PIG}-\text{KLE}}\) & \(\hat{\mathbb{a}}^{\text{PIG}-\text{MKLE}}\) \\ \hline \(x1\) & \(0.5436\) & \(0.0001\) & \(0.1434\) & \(0.0010\) & \(-0.0296\) & \(0.0430\) & \(-0.0001\) \\ \hline \(x2\) & \(0.4398\) & \(0.0002\) & \(0.1440\) & \(0.0008\) & \(0.2138\) & \(0.0348\) & \(-0.0002\) \\ \hline \(x3\) & \(-0.3814\) & \(-0.0002\) & \(-0.1318\) & \(-0.0007\) & \(-0.2544\) & \(-0.0302\) & \(0.0002\) \\ \hline \(x4\) & \(0.8503\) & \(0.0001\) & \(0.2116\) & \(0.0016\) & \(-0.1302\) & \(0.0672\) & \(-0.0001\) \\ \hline SMSE & \(987.4608\) & \(18.0435\) & \(17.5653\) & \(1.3558\) & \(171.9145\) & \(7.3136\) & \(1.3578\) \\ \hline PMSE & \(453.3143\) & \(446.1027\) & \(448.5152\) & \(446.1133\) & \(450.5399\) & \(446.6592\) & \(446.0973\) \\ \hline PMAE & \(16.1498\) & \(15.9001\) & \(15.9847\) & \(15.9005\) & \(16.0511\) & \(15.9198\) & \(15.8999\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Squirrel dataset estimated regression coefficients.