Asymptotic accuracy of the saddlepoint approximation for maximum likelihood estimation
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最大似然估计鞍点近似的渐近精度
- 研究问题
- 将鞍点密度近似作为似然后最大化，所得估计与真实最大似然估计相差多少，这种误差是否会影响统计推断？
- 方法与思路
- 作者把鞍点似然分解为精确项与近似项，分析对数似然梯度的近似误差，并在明确的可识别性条件下推导误差界。
- 材料陈述的贡献
- 摘要称不同参数的估计近似误差具有不同渐近阶，均小于推断不确定性；还给出简化鞍点近似及正态近似的相关误差结果。
- 阅读关注点
- 应核对可识别性条件、参数分组和渐近尺度，查阅原文确认摘要中压平的指数写法，并区分密度误差与估计量误差。
- 阅读边界
- 所给材料是稿件版本未注明的 OpenAlex 索引摘要，不能推断刊本变化；具体误差阶的指数排版及成立条件需查阅原文。
原文摘要
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/n2) for some parameters and O(1/n3/2) 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 approximation error when the saddlepoint approximation is replaced by the normal approximation.