Abstract
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 . 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 for some parameters and or 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 approximation error when the saddlepoint approximation is replaced by the normal approximation.
Funding Statement
This work was supported in part by funding from the Royal Society of New Zealand.
Acknowledgements
The author thanks Rachel Fewster and Joey Wei Zhang, whose use of saddlepoint MLEs sparked the author’s interest in this question; Rachel Fewster for helpful comments on drafts of this paper; and Godrick Oketch for suggesting the model of Example 26 in Appendix J as a useful one for explicit comparisons. Useful comments and questions from the anonymous referees helped to improve this paper.
Citation
Jesse Goodman. "Asymptotic accuracy of the saddlepoint approximation for maximum likelihood estimation." Ann. Statist. 50 (4) 2021 - 2046, August 2022. https://doi.org/10.1214/22-AOS2169
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