Asymptotic accuracy of the saddlepoint approximation for maximum likelihood estimation

The saddlepoint approximation gives an to the density of a random variable in terms its moment generating function. When underlying is itself sum n unobserved i.i.d. terms, basic classical result that relative error order 1/n. If instead interpreted as likelihood and maximised function model parameters, maximum estimate (MLE) can be much faster compute than true MLE. This paper proves analogous for between MLE MLE: subject certain explicit identifiability conditions, has asymptotic size O(1/n2) some parameters O(1/n3/2) or O(1/n) others. In all three cases, errors are asymptotically negligible compared inferential uncertainty. proof based on factorisation into exact approximate term, along with analysis gradient log-likelihood. also insight alternatives approximation, including new simpler which we derive bounds. As corollary our results, obtain when replaced by normal approximation.