  The parametric complexity is the key quantity in the minimum description
length (MDL) approach to statistical model selection. Rissanen and others have
shown that the parametric complexity of a statistical model approaches a simple
function of the Fisher information volume of the model as the sample size $n$
goes to infinity. This paper derives higher-order asymptotic expansions for the
parametric complexity, in the case of exponential families and independent and
identically distributed data. These higher-order approximations are calculated
for some examples and are shown to have better finite-sample behaviour than
Rissanen's approximation. The higher-order terms are given as expressions
involving cumulants (or, more naturally, the Amari-Chentsov tensors), and these
terms are likely to be interesting in themselves since they arise naturally
from the general information-theoretic principles underpinning MDL. The
derivation given here specializes to an alternative and arguably simpler proof
of Rissanen's result (for the case considered here), proving for the first time
that his approximation is $O(n^{-1})$.
