The log-likelihood of any exponential family is simply \(n[\bar{\s} \Tr \bt - \psi(\bt)]\) plus a constant, so
\[\as{ \u &\propto \bar{\s} - \nabla \psi(\bt) \\ \bth &= (\nabla\psi)^{-1}(\bar{\s}) }\]Since the MLE is simply a function of the mean in exponential families, it is particularly easy to derive its limiting distribution. Letting \(\bm = \Ex(\s)\), the central limit theorem tells us that
\[\sqrt{n}(\bar{\s} - \bm) \inD \Norm(\zero, \V),\]where \(\V = \nabla^2 \psi(\bt)\). Thus, letting \(\g\) denote the transformation \(\bt=\g(\bm)\), we have, by the delta method:
\[\sqrt{n}(\bth - \bt) \inD \Norm(\zero, \nabla \g (\bm) \Tr \V \nabla \g (\bm));\]keep in mind here that \(\nabla \g\) and \(\V\) are both \(d \times d\) matrices.
See also: Likelihood: Asymptotic normality.