The likelihood of $\bm{y}$, conditional on $\bm{\upsilon}$ is then
$$
\begin{align}
f_{\bm{y}|\bm{\upsilon}=\bm{\upsilon}}(\bm{y}, \bm{\upsilon}) &= \prod_{i=1}^{n_x} \left[ {\frac{1}{\left(2\pi\sigma^2\right)^{\frac{1}{2}}} \exp\left[-\frac{\left|\left|\bm{y}_i-\bm{X}_i\bm{\beta}-\bm{Z}_i\bm{\Lambda}(\bm{\theta})\bm{\upsilon} \right|\right|^2}{2\sigma^2}\right) } \right]^{\bm{\Omega}_{ii}} \\
&= \prod_{i=1}^{n_x} {\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\bm{\Omega}_{ii}}{2}}} \exp \left(- \frac{ \left|\left| \bm{\Omega}_{ii}^{\frac{1}{2}} \left(\bm{y}_i-\bm{X}_i\bm{\beta}-\bm{Z}_i\bm{\Lambda}(\bm{\theta})\bm{\upsilon} \right) \right|\right|^2}{2\sigma^2}\right) } \\
&= {\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii}}{2}}} \exp \left( - \frac{ \left|\left|\bm{\Omega}^{\frac{1}{2}} \left(\bm{y}-\bm{X}\bm{\beta}-\bm{Z}\bm{\Lambda}(\bm{\theta})\bm{\upsilon} \right) \right|\right|^2}{2\sigma^2}\right) }
\end{align}$$
And the unconditional density of $\bm{\upsilon}$ is
$$
\begin{align}
f_{\bm{\upsilon}}(\bm{\upsilon}) &=\prod_{j=1}^{n_z} \left[ \frac{1}{\left(2\pi\sigma^2\right)^{\frac{1}{2}}} \exp \left[- \frac{ \left|\left| \bm{\upsilon}_j \right|\right|^2}{2\sigma^2}\right] \right]^{\bm{\Psi}_{jj}} \\
&=\prod_{j=1}^{n_z} \frac{1}{\left(2\pi\sigma^2\right)^{\frac{\bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ \left|\left| \bm{\Psi}_{jj}^{\frac{1}{2}} \bm{\upsilon}_j \right|\right|^2}{2\sigma^2}\right] \\
&= \frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ \left|\left| \bm{\Psi}^{\frac{1}{2}} \bm{\upsilon} \right|\right|^2}{2\sigma^2}\right] \label{eq:WeMixWfu}
\end{align}$$
where $\sum \bm{\Psi}_{jj}$ is the sum of the terms in the diagonal matrix $\bm{\Psi}$.
The joint distribution of $\bm{\upsilon}$ and $\bm{y}$ is then the product of eqs. \ref{eq:WeMixWfyu} and \ref{eq:WeMixWfu}:
$$\begin{align}
f_{\bm{y}, \bm{\upsilon}} \left(\bm{y}, \bm{\upsilon}, \bm{\beta}, \bm{\theta}, \sigma^2 \right)&= f_{\bm{y}|\bm{\upsilon}=\bm{\upsilon}}(\bm{y}, \bm{\upsilon})\cdot f_{\bm{\upsilon}}(\bm{\upsilon})\\
&={\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii}}{2}}} \exp \left[- \frac{ \left|\left|\bm{\Omega}^{\frac{1}{2}} \left(\bm{y}-\bm{X\beta}-\bm{Z}\bm{\Lambda}(\bm{\theta})\bm{\upsilon} \right) \right|\right|^2}{2\sigma^2}\right] } \cdot \frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ \left|\left| \bm{\Psi}^{\frac{1}{2}} \bm{\upsilon} \right|\right|^2}{2\sigma^2}\right] \\
&=\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii} + \sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ \left|\left|\bm{\Omega}^{\frac{1}{2}} \left(\bm{y}-\bm{X\beta}-\bm{Z}\bm{\Lambda}(\bm{\theta})\bm{\upsilon} \right) \right|\right|^2 + \left|\left| \bm{\Psi}^{\frac{1}{2}} \bm{\upsilon} \right|\right|^2}{2\sigma^2} \right] \\
&=\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii} + \sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ r^2(\bm{\theta}, \bm{\beta}, \bm{\upsilon})}{2\sigma^2} \right]
\end{align}$$
Using the same logic for the results in eq. \ref{eq:r2sub}, $r^2$ can be written as a sum of the value at the optimum ($\hat{\bm{\beta}}$ and $\hat{\bm{\upsilon}}$) and deviations from that:
$$\begin{align}
f_{\bm{y}, \bm{\upsilon}} \left(\bm{y}, \bm{\upsilon}, \bm{\beta}, \bm{\theta}, \sigma^2 \right)&=\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii} + \sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ r^2(\bm{\theta}, \hat{\bm{\beta}}, \hat{\bm{\upsilon}}) - \left| \left| \bm{R}_{22}(\bm{\beta} - \hat{\bm{\beta}})\right| \right|^2 - \left| \left| \bm{R}_{11}(\bm{\upsilon} - \hat{\bm{\upsilon}}) + \bm{R}_{12}(\bm{\beta} - \hat{\bm{\beta}}) \right| \right|^2}{2\sigma^2} \right]
\end{align}$$
Now, finding the integral of this over $\bm{\upsilon}$,
$$\begin{align}
\mathcal{L}(\bm{\beta}, \bm{\theta}, \sigma^2; \bm{y})&=\int f_{\bm{y}, \bm{\upsilon}} \left(\bm{y}, \bm{\upsilon}, \bm{\beta}, \bm{\theta}, \sigma^2 \right) d\bm{\upsilon} \\
&=\int \frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii} + \sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ r^2(\bm{\theta}, \hat{\bm{\beta}}, \hat{\bm{\upsilon}}) - \left| \left| \bm{R}_{22}(\bm{\beta} - \hat{\bm{\beta}})\right| \right|^2 - \left| \left| \bm{R}_{11}(\bm{\upsilon} - \hat{\bm{\upsilon}}) + \bm{R}_{12}(\bm{\beta} - \hat{\bm{\beta}}) \right| \right|^2}{2\sigma^2} \right] d\bm{\upsilon} \\
&=\frac{1}{\left(2\pi\sigma^2\right)^{\frac{\sum_i \bm{\Omega}_{ii} + \sum_j \bm{\Psi}_{jj}}{2}}} \exp \left[- \frac{ r^2(\bm{\theta}, \hat{\bm{\beta}}, \hat{\bm{\upsilon}}) - \left| \left| \bm{R}_{22}(\bm{\beta} - \hat{\bm{\beta}})\right| \right|^2 }{2\sigma^2}\right] \int \exp \left[- \frac{\left| \left| \bm{R}_{11}(\bm{\upsilon} - \hat{\bm{\upsilon}}) + \bm{R}_{12}(\bm{\beta} - \hat{\bm{\beta}}) \right| \right|^2}{2\sigma^2} \right] d\bm{\upsilon} \label{eq:tmpLA}
\end{align}$$
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