Saddlepoint Approximation

Since no closed-form expression for the joint PDF of $ {\bf z}$ in (8.1) is known, we apply the Saddlepoint approximation [47],[16]. To obtain the SPA, we need the joint cumulant generating function (CGF) of $ {\bf z}$, namely,

$\displaystyle c_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) \stackrel{\mbox{\tiny$\Delta$}}{=}\log g_z(\mbox{\boldmath$\lambda$}),
$

where $ g_z($$ \lambda$$ )$ is the joint moment-generating function (MGF) of $ {\bf z}$. Also, we need the first and second-order partial derivatives of $ c_z($$ \lambda$$ )$. Once these are known, the formulas in reference [16] may be used to obtain the SPA.

It is shown in [48] that

$\displaystyle c_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) = -\frac{1}{2} \log \left\vert{\bf Q}(\mbox{\boldmath$\lambda$}...
...h$\lambda$}) {\bf t}(\mbox{\boldmath$\lambda$}) + u(\mbox{\boldmath$\lambda$}),$ (8.5)

where

$\displaystyle {\bf Q}($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) = {\bf I}_M - 2 {\bf D}($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ),$

with

$\displaystyle {\bf D}($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) \stackrel{\mbox{\tiny$\Delta$}}{=}\sum_{m=1}^M
\lambda_m {\bf ...
... \stackrel{\mbox{\tiny$\Delta$}}{=}\sum_{m=1}^M
\lambda_m {\bf p}_m, \;\;\;\;
$

and

$\displaystyle u($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) \stackrel{\mbox{\tiny$\Delta$}}{=}\sum_{m=1}^M
\lambda_m q_m.
$

The first-order partial derivatives are

$\displaystyle \begin{array}{rcl}
{\partial \over \partial \lambda_m} \; c_z(\m...
...Q}^{-1} {\bf t}
+ {\bf t}^\prime {\bf B}_m
{\bf Q}^{-1} {\bf t},
\end{array}$

for $ 1\leq m \leq M$, and the second-order partial derivatives are

$\displaystyle \begin{array}{rcl}
{\partial^2 \over \partial
\lambda_l \partia...
...\\
&& + 4{\bf t}^\prime {\bf B}_l {\bf B}_m {\bf Q}^{-1} {\bf t}
\end{array}$

where

$\displaystyle {\bf B}_m($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) \stackrel{\mbox{\tiny$\Delta$}}{=}
{\bf Q}^{-1}(\mbox{\boldmath$\lambda$}) {\bf P}_m,
$

and we drop the $ ($$ \lambda$$ )$ dependence from $ {\bf t}($$ \lambda$$ )$, $ {\bf Q}^{-1}($$ \lambda$$ )$, and $ {\bf B}_m($$ \lambda$$ )$, for simplicity. The third and fourth derivatives, necessary for the first-order correction term of the SPA have also been worked out [48].

The equations simplify considerably if we assume that $ \{{\bf p}_m\}$ and $ \{q_m\}$ are all zero and compute the PDF under the WGN assumption $ H_0$. We then have

$\displaystyle c_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) = \log g_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) = -\frac{1}{2} \log \left\vert{\bf Q}(\mbox{\boldmath$\lambda$})\right\vert.$ (8.6)

The first order partial derivatives reduce to

$\displaystyle \begin{array}{rcl}
{\partial \over \partial \lambda_m} \; c_z(\mbox{\boldmath$\lambda$})
& = &
{\rm tr}\left\{ {\bf B}_m \right\}
\end{array}$

and the second order partial derivatives become

$\displaystyle \begin{array}{rcl}
{\partial^2 \over \partial
\lambda_l \partia...
...rm tr}\left\{ {\bf B}_l
{\bf B}_m \right\}, \;\; 1\leq l,m\leq M.
\end{array}$

The SPA algorithm is provided in software/pdf_quadspa.m, which assumes that $ \{{\bf p}_m\}$ and $ \{q_m\}$ are all zero and computed the PDF under the WGN ($ H_0$) assumption.

Baggenstoss 2017-05-19