## 4 Simulations
### Hypothesis Testing  
In this subsection, we explore the performances of the likelihood ratio, Wald and score statistics presented in (18)-(20) for the hypothesis testing in (17) with various parameter configurations. The sample sizes are set to be \(n=50(50)400\), where \(n_{1}(s)n_{2}\) means from \(n_{1}\) to \(n_{2}\) with step size \(s\), and other parameters are set as follows:  
1. When \(q=2\), \(\mathbf{\beta}=(\beta_{0},\beta_{1})^{\top}\), \(\alpha\) is set to be \(0.75,\ 0.85,\ 0.95\), \(\mathbf{C}=(0\ 1\ 0)\), \(\mathbf{c}_{r}=0\) and \(\mathbf{\theta}=(\mathbf{\beta}^{\top},\mathbf{\alpha})^{\top}\), so that (17) becomes \(H_{0}\): \(\beta_{1}=0\). The true value of \(\mathbf{\beta}\) in \(H_{0}\) is \(\mathbf{\beta}=(1,0)^{\top}\), while the value of \(\mathbf{\beta}\) in \(H_{1}\) is \(\mathbf{\beta}=(1,0.5)^{\top}\). We generate \(\{w_{i1}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}N(0.1,0.2)\) and set \(\mathbf{w}_{i}=(1,w_{i1})^{\top}\);
2. When \(q=4\), \(\mathbf{\beta}=(\beta_{0},\beta_{1},\beta_{2},\beta_{3})^{\top}\), \(\alpha\) is set to be \(0.75,\ 0.85,\ 0.95\), \[\mathbf{C}=\left( \begin{array}{cccccc}0&1&-1&0&0\\ 0&0&1&-1&0\\ 0&0&0&1&0\\ \end{array} \right),\quad\mathbf{c}_{r}=\mathbf{0}_{3}\quad\text{and}\quad\mathbf{\theta}=( \mathbf{\beta}^{\top},\mathbf{\alpha})^{\top},\] so that (17) becomes \(H_{0}\): \(\beta_{1}=\beta_{2}=\beta_{3}=0\). The true value of \(\mathbf{\beta}\) in \(H_{0}\) is \(\mathbf{\beta}=(1,0,0,0)^{\top}\) and the value of \(\mathbf{\beta}\) in \(H_{1}\) is \(\mathbf{\beta}=(1,-1,1,-0.5)^{\top}\). We generate \(\{w_{i1}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}N(0.1,0.05)\), \(\{w_{i2}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}U(0,0.1)\), \(\{w_{i3}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}0.4\times \text{Bernoulli}(0.5)\), and set \(\mathbf{w}_{i}=(1,w_{i1},w_{i2},w_{i3})^{\top}\);
3. When \(q=2\), \(\mathbf{\beta}=(1,1)^{\top}\), \(\mathbf{C}=(0\ 0\ 1)\), \(\mathbf{c}_{r}=1\) and \(\mathbf{\theta}=(\mathbf{\beta}^{\top},\mathbf{\alpha})^{\top}\), so that (17) becomes \(H_{0}\): \(\alpha=1\). The alternative values of \(\alpha\) in \(H_{1}\) are set as \(0.9\) and \(0.95\). We generate \(\{w_{i1}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}N(1,0.1)\) and set \(\mathbf{w}_{i}=(1,w_{i1})^{\top}\);
4. When \(q=4\), \(\mathbf{\beta}=(1,-1,1,-0.5)^{\top}\), \(\mathbf{C}=(0\ 0\ 0\ 0\ 1)\), \(\mathbf{c}_{r}=1\) and \(\mathbf{\theta}=(\mathbf{\beta}^{\top},\mathbf{\All hypothesis testings are conducted at a significant level of 0.05. To calculate the empirical levels of the three tests, we first generate \(\{Y_{i}=y_{i}\}_{i=1}^{\text{ind}}\overset{\text{GP}}{\sim}\text{GP}^{(0)}(w_{i}^ {\prime}\mathbf{\beta},\alpha)\) under \(H_{0}\). Repeating this process for \(L\) (=10,000) times, we obtained \(Y_{\text{obs}}^{(l)}=\{y_{1}^{(l)},\ldots,y_{n}^{(l)}\}_{i=1}^{L}\). Since our MM algorithm (14) & (16) is designed for \(\alpha<1\), we apply two-stage method to obtain the MLEs of \(\mathbf{\theta}\) for the \(\text{GP}^{(l)}\) regression model. In the first stage, we calculate the MLEs \(\{\mathbf{\hat{\beta}},\hat{\alpha}\}\) via the MM algorithm (14) & (16) with the generated \(\{y_{i}\}_{i=1}^{n}\) and corresponding covariate vectors \(\{\mathbf{w}_{i}\}_{i=1}^{n}\). If the estimated \(\hat{\alpha}<1\), implying that the dataset is under-dispersed, we shall keep the estimation result and will not go to the second stage. If the estimated \(\hat{\alpha}=1\), implying that the dataset may be equal- or over-dispersion, we shall go to the next stage; that is recalculating the MLEs \(\{\mathbf{\hat{\beta}},\hat{\alpha}\}\) through the '\(\text{vglm}\)' function by choosing family as 'genpoisson1' in \(\text{VGAM}\ R\) package because this function can only calculate the MLEs of the parameter when \(\alpha\geqslant 1\). Let \(\{r_{j}\}_{j=1}^{3}\) denote the number of rejecting the null hypothesis \(H_{0}\) by the likelihood ratio, Wald and score statistics, respectively. Hence, the actual significance level can be estimated by \(r_{j}/L\) under \(H_{0}\).