Adapt latex from stan-math doxygen

Author: WardBrian

src/functions-reference/positive_lower-bounded_distributions.qmd

@@ -159,8 +159,32 @@ For a description of argument and return types, see section
 
 ### Probability density function
 
-If $\alpha \in \mathbb{R}^+$, $\tau \in \mathbb{R}^+$, $\beta \in [0,
-1]$ and $\delta \in \mathbb{R}$, then for $y > \tau$, \begin{equation*}
+If $\alpha \in \mathbb{R}^+$, $\tau \in \mathbb{R}^+$, $\beta \in [0, 1]$,
+$\delta \in \mathbb{R}$, $s_{\delta} \in \mathbb{R}^{\geq 0}$, $s_{\beta} \in [0, 1)$, and
+$s_{\tau} \in \mathbb{R}^{\geq 0}$ then for $y > \tau$,
+
+\begin{eqnarray*}
+\text{Wiener}(y|\alpha, \tau, \beta, \delta, s_{\delta}, s_{\beta}, s_{\tau})
+
+&=& \frac{1}{s_{\tau}}
+\int_{\tau}^{\tau + s_{\tau}} \frac{1}{s_{\beta}}\int_{\beta -0.5s_{\beta}}^{\beta + 0.5s_{\beta}}
+M \times p_3(y-\tau|a,\delta,\omega) \ d\omega \ d\tau,
+\end{eqnarray*}
+where $M$ and $p_3()$ are defined, by using $t:=y-\tau$, as
+ \begin{eqnarray*}
+ M := \frac{1}{\sqrt{1+s^2_{\delta} t}}
+ \mathbb{e}^{a\delta\omega+\frac{\delta^2t}{2}+\frac{s^2_{\delta} a^2
+ \omega^2-2a\delta\omega-\delta^2t}{2(1+s^2_{\delta}t)}} \text{ and} \\ p_3(t|a,\delta,\beta) :=
+ \frac{1}{a^2} \mathbb{e}^{-a \delta \beta -\frac{\delta^2t}{2}} f(\frac{t}{a^2}|0,1,\beta), \end{eqnarray*}
+ where  $f(t^*=\frac{t}{a^2}|0,1,\beta)$ has two forms
+ \begin{eqnarray*}
+ f_l(t^*|0,1,\beta) = \sum_{k=1}^{\infty} k\pi \mathbb{e}^{-\frac{k^2\pi^2t^*}{2}}
+ \sin{(k \pi \beta)}\text{ and} \\
+ f_s(t^*|0,1,\beta) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi (t^*)^3}}
+ (\beta+2k) \mathbb{e}^{-\frac{(\beta+2k)^2}{2t^*}}, \end{eqnarray*}
+
+In the case where $s_{\delta}$, $s_{\beta}$, and $s_{\tau}$ are all $0$, this simplifies to
+\begin{equation*}
 \text{Wiener}(y|\alpha, \tau, \beta, \delta) =
 \frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta -
 \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k +