## 2 Methodology
### Biased Estimators  
When multicollinearity is found in the data, corrective measures ought to be implemented to mitigate its effects. Multicollinearity poses a serious threat to the estimates of the regression coefficients [28; 29], as it can lead to inflated variances of the estimated parameters. Consequently, the precision of these estimates is reduced. One approach to addressing multicollinearity is removing highly correlated regressors. However, an alternative is to retain all regressors and employ a biased estimator rather than the maximum likelihood estimator, which may offer advantages in certain contexts.  
Biased estimators have been proposed to increase the accuracy of the parameter estimate in the model when multicollinearity exists, especially when the parameter estimation is the focus of the regression analysis. Following the work of [PERSON] and [PERSON] (2007), [PERSON] et al. (2016) extended the James-Stein estimator to the Poisson-Inverse-Gaussian (PIG) regression model as:  
\[\hat{\beta}^{PIG-ISE}=c\hat{\varrho}^{PIG-MLE},\ \ \ \ \ 0<c<1 \tag{9}\]  
where  
\[c=\frac{\hat{\beta}^{PIG-MLE}\hat{\beta}^{PIG-MLE}}{\hat{\beta}^{PIG-MLE^{ \prime}}\hat{\beta}^{PIG-MLE}+\hat{\psi}\text{tr}\big{(}\text{cov}\big{(}\hat{ \beta}^{PIG-MLE}\big{)}\big{)}}.\]  
The MMSE and SMSE for \(\hat{\beta}^{PIG-ISE}\) are defined as follows:  
\[MMSE\big{(}\hat{\alpha}^{PIG-JSE}\big{)}=c^{2}\hat{\omega}\Lambda^{-1}+(c-1) \alpha^{\prime}\alpha(c-1) \tag{10}\]  
\[SMSE\big{(}\hat{\alpha}^{PIG-JSE}\big{)}=c^{2}\hat{\omega}{\sum_{j=1}^{q}} \frac{1}{\lambda_{j}}+(c-1)^{2}{\sum_{j=1}^{q}}\alpha_{j}^{2} \tag{11}\]  
[PERSON] and [PERSON] (2007) developed the ridge estimator to address multicollinearity in the linear regression model. [PERSON] et al. (2016) extended the ridge regression estimator to the Poisson-Inverse-Gaussian (PIG) regression model as:  
\[\hat{\beta}^{PIG-RRE}=\Lambda_{k^{+}}^{-1}(X^{\prime}\hat{W}X)\hat{\beta}^{ PIG-MLE},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k>0 \tag{12}\]  
where \(\Lambda_{k^{+}}^{-1}=\big{(}X^{\prime}\hat{W}X+kI\big{)}^{-1}\).  
The MMSE and SMSE for \(\hat{\beta}^{PIG-RRE}\) are defined as follows:  
\[MMSE\big{(}\hat{\alpha}^{PIG-RRE}\big{)}=\hat{\omega}\Lambda_{k^{+}}^{-1} \Lambda\Lambda_{k^{+}}^{-1}+k^{2}\Lambda_{k^{+}}^{-1}\alpha^{\prime}\alpha \Lambda_{k^{+}}^{-1} \tag{13}\]  
\[SMSE\big{(}\hat{\alpha}^{PIG-RRE}\big{)}=\hat{\omega}{\sum_{j=1}^{q}}\frac{ \lambda_{j}}{\big{(}\lambda_{j}+k\big{)}^{2}}+k^{2}{\sum_{j=1}^{q}}\frac{ \alpha_{j}^{2}}{\big{(}\lambda_{j}+k\big{)}^{2}} \tag{14}\]  
Following [PERSON] and [PERSON] (2007), [PERSON] (2007) introduced a biased estimator that competes favorably with the ridge at reducing the impact of collinearity. [PERSON] et al. (2016) proposed the adjusted Liu estimator, which is defined as follows:  
\[\hat{\beta}^{PIG-LE}=\Lambda_{I}^{-1}\Lambda_{d^{+}}\hat{\beta}^{PIG-MLE}\  
Based on the research of [PERSON] and [PERSON] [30] and [PERSON] et al. [31], [PERSON] [32] introduced a new method for the PIG regression model with two parameters as follows:  
\[\hat{\beta}^{PIG-TPE}=\Lambda_{k^{+}}^{-1}\Lambda_{kl}\hat{\beta}^{PIG-MLE}\qquad \qquad\quad 0\leq d<1,\ 0<k \tag{18}\]  
where \(\Lambda_{kl}=\left(X^{\prime}\hat{W}X+kdI\right)\). Hence, the MMSE and SMSE for \(\hat{\beta}^{PIG-TPE}\) are defined as follows:  
\[MMSE\left(\hat{\alpha}^{PIG-TPE}\right)=\hat{\omega}\Lambda_{k^{+}}^{-1} \Lambda_{kl}\Lambda^{-1}\Lambda_{kl}\Lambda_{k^{+}}^{-1}+(d-1)^{2}k^{2}\Lambda_ {k^{+}}^{-1}\alpha^{\prime}\alpha\Lambda_{k^{+}}^{-1} \tag{19}\]  
\[SMSE\left(\hat{\alpha}^{PIG-TPE}\right)=\hat{\omega}{\sum_{j=1}^{q}}\frac{ \left(\lambda_{j}+kd\right)^{2}}{\left(\lambda_{j}+k\right)^{2}\lambda_{j}}+( d-1)^{2}k^{2}{\sum_{j=1}^{q}}\frac{\alpha_{j}^{2}}{\left(\lambda_{j}+k \right)^{2}}. \tag{20}\]