and $\lim \sup |C|<\infty$. Furthermore, suppose that is replaced by
$$
\left.\Sigma^{-1 / 2} G^{-1} \frac{\partial l}{\partial \psi}\right|{\psi{0}} \longrightarrow N(0, I) \text { in distribution, }
$$
where ${\Sigma}$ is a sequence of positive definite matrices such that
$$
0<\liminf \lambda_{\min }(\Sigma) \leq \lim \sup \lambda_{\max }(\Sigma)<\infty
$$
and $I$ is the $p$-dimensional identity matrix. Then, the asymptotic distribution of $\mathcal{W}$ is $\chi_{q-p}^{2}$
The proofs are given in . According to the proof, one has $G[\hat{\psi}-\psi(\hat{\phi})]=O_{P}(1)$, hence
$$
\begin{aligned}
\hat{\mathcal{W}} &=[\hat{\theta}-\theta(\hat{\phi})]^{\prime} G\left[Q_{w}^{-}+o_{P}(1)\right] G[\hat{\theta}-\theta(\hat{\phi})] \
&=\mathcal{W}+o_{P}(1)
\end{aligned}
$$
Thus,we conclude the following.
STAT4022 COURSE NOTES ：
where $b(\cdot), a_{i}(\cdot), c_{i}(\cdot, \cdot)$ are known functions, and $\phi$ is a dispersion parameter which may or may not be known. The quantity $\xi_{i}$ is associated with the conditional mean $\mu_{i}=\mathrm{E}\left(y_{i} \mid \alpha\right)$, which, in turn, is associated with a linear predictor
$$
\eta_{i}=x_{i}^{\prime} \beta+z_{\mathrm{i}}^{\prime} \alpha,
$$
where $x_{i}$ and $z_{i}$ are known vectors and $\beta$ a vector of unknown parameters (the fixed effects), through a known link function $g(\cdot)$ such that
$$
g\left(\mu_{i}\right)=\eta_{i}
$$
Furthermore, it is assumed that $\alpha \sim N(0, G)$, where the covariance matrix $G$ may depend on a vector $\theta$ of unknown variance components.
Note that, according to the properties of the exponential family, one has $b^{\prime}\left(\xi_{i}\right)=\mu_{i}$. In particular, under the so-called canonical link, one has
$$
\xi_{i}=\eta_{i} ;
$$