## 4 Simulations
### Hypothesis Testing
(**a**) The empirical power with \(H_{1}\): \(\mathbf{\beta}=(1,-1,1,-0.5)^{\top}\) for \(\alpha=0.9\); (**b**) The empirical power with \(H_{1}\): \(\mathbf{\beta}=(1,-1,1,-0.5)^{\top}\) for \(\alpha=0.95\); (**c**) The empirical significant level with \(H_{1}\): \(\mathbf{\beta}=(1,-1,1,-0.5)^{\top}\) for \(\alpha=1\). Comparisons of the GP\({}^{(1)}\) Regression Model with the Conway-Maxwell-Poisson Regression Model  
To compare the performance of the goodness-of-fit tests and computational complexity in the GP\({}^{(1)}\) regression model (3) and the _Conway-Maxwell-Poisson_ (CMP) regression model, we consider using both models to fit a dataset, which is generated from one of the two models. A discrete r.v. \(Y\) is said to follow the CMP distribution with parameters \(\lambda>0\) and \(\
u>0\), denoted by \(Y\sim\text{CMP}(\lambda,\
u)\), if its pmf is [16]:  
\[\Pr(Y=y) = \frac{\lambda^{y}}{(y!)^{\
u}Z(\lambda,\
u)},\quad y=0,1,2,\ldots,\infty,\]  
where \(Z(\lambda,\
u)=\sum_{s=0}^{\infty}\lambda^{s}/(s! )^{\
u}\) is a normalizing constant and \(\
u\) is the dispersion parameter. The CMP\((\lambda,\
u)\) distribution reduces to the Poisson\((\lambda)\) when \(\
u=1\), and it has the twin properties of over-dispersion when \(\
u<1\) and under-dispersion when \(\
u>1\). The CMP regression model [3,17] is:  
\[\{Y_{i}\}_{i=1}^{n}\stackrel{{\text{ind}}}{{\sim}}\text{CMP}( \lambda_{i},\
u)\quad\text{and}\quad\log(\lambda_{i})=\mathbf{w}_{i}^{\top}\mathbf{ \beta},\quad i=1,\ldots,n.\]  
The sample size is set to be \(n=1000\), \(q=4\), \(\mathbf{\beta}=(1,-0.5,1,0.5)^{\top}\), \(\mathbf{w}_{i}=(1,w_{i1},w_{i2},w_{i3})^{\top}\), \(\{w_{i1}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}N(0.1,0.5)\), \(\{w_{i2}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}U(0,1)\), \(\{w_{i3}\}_{i=1}^{n}\stackrel{{\text{iid}}}{{\sim}}\text{Bernoulli}(0.5)\) and other parameter configurations are set as follows:  
* For a fixed \(\mathbf{\beta}\), set \(\alpha=0.9\) and generate \(X_{i}=x_{i}\stackrel{{\text{ind}}}{{\sim}}\text{GP}^{(0)}(\mu_{i},\alpha)\) with \(\mu_{i}=\exp(\mathbf{w}_{i}^{\top}\mathbf{\beta})\) for \(i=1,\ldots,n\). * For a fixed \(\mathbf{\beta}\), set \(\
u=1.2\) and generate \(X_{i}=x_{i}\stackrel{{\text{ind}}}{{\sim}}\text{CMP}(\lambda_{i},\
u)\) with \(\lambda_{i}=\exp(\mathbf{w}_{i}^{\top}\mathbf{\beta})\) for \(i=1,\ldots,n\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{\(n\)} & \multicolumn{4}{c}{\(\mathbf{\alpha=0.9}\)} & \multicolumn{4}{c}{\(\mathbf{\alpha=0.95}\)} & \multicolumn{4}{c}{\(\mathbf{\alpha=1}\)} \\ \cline{2-11}  & \(T_{L}\) & \(T_{W}\) & \(T_{S}\) & \(T_{L}\) & \(T_{W}\) & \(T_{S}\) & \(T_{L}\) & \(T_{W}\) & \(T_{S}\) \\ \hline
50 & 0.1652 & 0.2764 & 0.0737 & 0.0881 & 0.1542 & 0.0445 & 0.0603 & 0.0884 & 0.0494 \\
100 & 0.2553 & 0.3641 & 0.1616 & 0.0999 & 0.1548 & 0.0608 & 0.0555 & 0.0672 & 0.0479 \\
150 & 0.3562 & 0.4563 & 0.2642 & 0.1203 & 0.1725 & 0.0834 & 0.0525 & 0.0614 & 0.0475 \\
200 & 0.4738 & 0.5604 & 0.3814 & 0.1426 & 0.1929 & 0.0993 & 0.0533 & 0.0601 & 0.0498 \\
250 & 0.5652 & 0.6446 & 0.4870 & 0.1689 & 0.2152 & 0.1286 & 0.0523 & 0.0587 & 0.0505 \\
300 & 0.6560 & 0.7207 & 0.5876 & 0.1901 & 0.2423 & 0.1500 & 0.0516 & 0.0543 & 0.0500 \\
350 & 0.7272 & 0.7779 & 0.6710 & 0.1996 & 0.2503 & 0.1623 & 0.0546 & 0.0583 & 0.0520 \\
400 & 0.7891 & 0.8332 & 0.7418 & 0.2286 & 0.2765 & 0.1886 & 0.0516 & 0.0573 & 0.0496 \\ \hline \end{tabular}
\end{table} 
Table 7: The empirical levels/powers of statistics (\(T_{L}\), \(T_{W}\), \(T_{S}\)) for Case (A3).