  The saddlepoint approximation gives an approximation to the density of a
random variable in terms of its moment generating function. When the underlying
random variable is itself the sum of $n$ unobserved i.i.d. terms, the basic
classical result is that the relative error in the density is of order $1/n$.
If instead the approximation is interpreted as a likelihood and maximised as a
function of model parameters, the result is an approximation to the maximum
likelihood estimate (MLE) that can be much faster to compute than the true MLE.
This paper proves the analogous basic result for the approximation error
between the saddlepoint MLE and the true MLE: subject to certain explicit
identifiability conditions, the error has asymptotic size $O(1/n^2)$ for some
parameters, and $O(1/n^{3/2})$ or $O(1/n)$ for others. In all three cases, the
approximation errors are asymptotically negligible compared to the inferential
uncertainty.
  The proof is based on a factorisation of the saddlepoint likelihood into an
exact and approximate term, along with an analysis of the approximation error
in the gradient of the log-likelihood. This factorisation also gives insight
into alternatives to the saddlepoint approximation, including a new and simpler
saddlepoint approximation, for which we derive analogous error bounds. As a
corollary of our results, we also obtain the asymptotic size of the MLE error
approximation when the saddlepoint approximation is replaced by the normal
approximation.
