To obtain the SPA to the PDF of , we need
the joint cumulant generating function (CGF)
of and its partial derivatives. The joint CGF is defined by

where
**log** is the vector function
which operates on each element of its argument and
is the vector sum,
which adds up all the elements of the argument.
If we define
as the element-by-element
reciprocal of
** **,
and
as the diagonal
-by- matrix with elements
equal to the elements of
** **,
it is straight-forward to show that the gradient vector
of
is the -by-1 vector

and the -by- Hessian matrix of
is

Baggenstoss
2017-05-19