Saddlepoint Approximation

Since no closed-form expression for the joint PDF of ${\bf z}$ in (9.1) is known, we apply the Saddlepoint approximation [55],[16]. To obtain the SPA, we need the joint cumulant generating function (CGF) of ${\bf z}$, namely,

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

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 [56] that

$$c_z( \mbox{\boldmath$\lambda$} ) = -\frac{1}{2} \log \left\vert{\bf Q}(\mbox{\boldmath$\lambda$}... ...h$\lambda$}) {\bf t}(\mbox{\boldmath$\lambda$}) + u(\mbox{\boldmath$\lambda$}),$$ (9.5)

where

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

with

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

and

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

The first-order partial derivatives are

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

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

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

where

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

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 [56].

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

$$c_z( \mbox{\boldmath$\lambda$} ) = \log g_z( \mbox{\boldmath$\lambda$} ) = -\frac{1}{2} \log \left\vert{\bf Q}(\mbox{\boldmath$\lambda$})\right\vert.$$ (9.6)

The first order partial derivatives reduce to

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

and the second order partial derivatives become

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

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.