## 4 Simulations  
### Accuracy of MLEs of Parameters  
To investigate the accuracy of MLEs of parameters, we consider dimensions: \(q=2,4\). The sample sizes are set to be \(n=100,200,400\); \(\alpha=0.6,0.8,0.95\) and other parameters are set as follows:  
1. When \(q=2\), \(\mathbf{\beta}=(1,-1)^{\top}\); \(\mathbf{w}_{i}=(1,w_{i1})^{\top}\), \(\{w_{i1}\}_{i=1}^{n}\stackrel{{\mathrm{iid}}}{{\sim}}N(0.3,\sigma _{0}^{2})\) with \(\sigma_{0}^{2}=0.5\);
2. When \(q=4\), \(\mathbf{\beta}=(1,-1,2,-2)^{\top}\); \(\mathbf{w}_{i}=(1,w_{i1},w_{i2},w_{i3})^{\top}\), \(\{w_{i1}\}_{i=1}^{n}\stackrel{{\mathrm{iid}}}{{\sim}}N(0.3,0.5)\), \(\{w_{i2}\}_{i=1}^{n}\stackrel{{\mathrm{iid}}}{{\sim}}U(0,1)\), \(\{w_{i3}\}_{i=1}^{n}\stackrel{{\mathrm{iid}}}{{\sim}}\mathrm{ Bernoulli}(0.5)\).  
For a given \(\{q,n,\mathbf{\beta},\alpha\}\), we first generate \(\{\mathbf{w}_{i}\}_{i=1}^{n}\), and then generate \(\{Y_{i}=y_{i}\}_{i=1}^{n}\stackrel{{\mathrm{iid}}}{{\sim}}\mathrm{ GP}^{(l)}(\mathbf{w}_{i}^{\top}\mathbf{\beta},\alpha)\) by the inversion method [15] based on the pmf given by (2). Then, we can calculate the MLEs \(\{\hat{\mathbf{\beta}},\hat{\alpha}\}\) via the MM algorithm (14) and (16) with the generated \(\{y_{i}\}_{i=1}^{n}\) and corresponding covariate vectors \(\{\mathbf{w}_{i}\}_{i=1}^{n}\). Finally, we independently repeat this process 10,000 times.  
The resultant average bias (denoted by Bias; i.e., average MLE minus the true value of the parameter) and the mean square error (denoted by MSE; i.e., Bias\({}^{2}\) + (standard deviation)\({}^{2}\), the standard deviation is estimated by the sample standard deviation of 10,000 MLEs) are reported in Tables 1 and 2.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{**Para**} & \multicolumn{2}{c}{\(\mathbf{\alpha=0.6}\)} & \multicolumn{2}{c}{\(\mathbf{\alpha=0.8}\)} & \multicolumn{2}{c}{\(\mathbf{\alpha=0.95}\)} \\ \cline{3-8}  & & **Bias** & **MSE** & **Bias** & **MSE** & **Bias** & **MSE** \\ \hline \multirow{3}{*}{100} & \(\beta_{0}\) & \(-0.0017\) & \(0.0371\) & \(-0.0034\) & \(0.0494\) & \(-0.0044\) & \(0.0582\) \\  & \(\beta_{1}\) & \(-0.0011\) & \(0.0703\) & \(-0.0001\) & \(0.0966\) & \(-0.0003\) & \(0.1186\) \\  & \(\alpha\) & \(-0.0085\) & \(0.0394\) & \(-0.0100\) & \(0.0535\) & \(-0.0187\) & \(0.0556\) \\ \hline \multirow{3}{*}{200} & \(\beta_{0}\) & \(-0.0007\) & \(0.0260\) & \(-0.0007\) & \(0.0350\) & \(-0.0020\) & \(0.0415\) \\  & \(\beta_{1}\) & \(0.0003\) & \(0.0543\) & \(-0.0001\) & \(0.0764\) & \(-0.0001\) & \(0.0916\) \\  & \(\alpha\) & \(-0.0037\) & \(0.0280\) & \(-0.0051\) & \(0.0373\) & \(-0.0084\) & \(0.0417\) \\ \hline \multirow{3}{*}{400} & \(\beta_{0}\) & \(-0.0004\) & \(0.0183\) & \(-0.0003\) & \(0.0242\) & \(-0.0009\) & \(0.0292\) \\  & \(\beta_{1}\) & \(0.0004\) & \(0.0357\) & \(0.0000\) & \(0.0484\) & \(-0.0006\) & \(0.0589\) \\ \cline{1-1}  & \(\alpha\) & \(-0.0014\) & \(0.0202\) & \(-0.0028\) & \(0.0267\) & \(-0.0042\) & \(0.0314\) \\ \hline \end{tabular}
\end{table}
Table 1: Parameter estimates based on 10,000 replications for Case (A1).  
Tables 1 and 2 showed that the absolute values of Bias and MSE tend to zero with the growth of data size for each parameter in Cases (A1) and (B1). For fixing else parameters, the absolute values of Bias and MSE are small for a small \(\alpha\).