Joint MGF of $ {\bf z}$.

The joint MGF of $ {\bf z}$ is, for $ \lambda$$ =[\lambda_1 \lambda_2 \ldots \lambda_M]^\prime$,

\begin{displaymath}\begin{array}{rcl} g_z(\mbox{\boldmath$\lambda$}) & \stackrel...
...\lambda_{M-1} y_{t_{M-1}} + \lambda_M r) \right\} . \end{array}\end{displaymath} (7.10)

This can be written

$\displaystyle g_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) = g_0({\bf A}$$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle )
$

where $ {\bf A}$ is the $ N$-by-$ M$ matrix that has 1's everywhere in the $ M$-th column except for 0's in rows $ t_1, t_2 \ldots t_{M-1}$, and $ {\bf A}$ has 1's in row $ t_1$, column 1; row $ t_2$, column 2; etc. Therefore, from (7.9),

$\displaystyle g_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) =
{1 \over {\rm prod}({\underline{\bf 1}}- {\bf Q} {\bf A}\mbox{\boldmath$\lambda$})}.
$

Note that $ {\bf Q}$ can be a large $ N$-by-$ N$ matrix if $ N$ is large. However, if we define

$\displaystyle {\bf P}\stackrel{\mbox{\tiny$\Delta$}}{=}{\bf Q}{\bf A},
$

$ {\bf P}$ is a reasonable $ N$-by-$ M$ size matrix and can be easily formed directly. The final simplified form for the joint MGF is

$\displaystyle g_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) =
{1 \over {\rm prod}({\underline{\bf 1}}- {\bf P}\mbox{\boldmath$\lambda$})}.
$



Baggenstoss 2017-05-19