论文标题
鞍点近似的渐近精度以进行最大似然估计
Asymptotic accuracy of the saddlepoint approximation for maximum likelihood estimation
论文作者
论文摘要
鞍点的近似在其力矩生成函数方面给出了随机变量的密度的近似值。当基本随机变量本身是$ n $未观察到的i.i.d.的总和。术语，基本的经典结果是密度的相对误差为$ 1/n $。相反，如果将近似值解释为模型参数的函数，则结果是对最大似然估计（MLE）的近似值，而计算的速度比真实的MLE要快得多。本文证明了鞍点MLE与真实MLE之间的近似误差的类似基本结果：在某些明确的可识别性条件下，该误差的某些参数具有渐近尺寸$ O（1/N^2）$，而$ O（1/N^{3/2}}）$（1/n^{3/2}）$（1/n^{3/2}）$（1/N）$（1/n）$（1/n）$（1/n）$。在所有三种情况下，与推论不确定性相比，近似误差在渐近上可以忽略不计。 该证明是基于将鞍点的可能性分解为精确和近似项的分解，以及对数可能的梯度中近似误差的分析。这种分解还可以深入了解鞍点近似的替代方案，包括一个新的，更简单的鞍点近似，我们得出类似的误差界限。作为我们结果的推论，当鞍点近似被正常近似代替时，我们还获得了MLE误差近似的渐近尺寸。
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.
