We present a complete framework for determining the asymptotic (or logarithmic) efficiency of estimators of large deviation probabilities and rate functions based on importance sampling. The framework relies on the idea that importance sampling in that context is fully characterized by the joint large deviations of two random variables: the observable defining the large deviation probability of interest and the likelihood factor (or Radon–Nikodym derivative) connecting the original process and the modified process used in importance sampling. We recover with this framework known results about the asymptotic efficiency of the exponential tilting and obtain new necessary and sufficient conditions for a general change of process to be asymptotically efficient. This allows us to construct new examples of efficient estimators for sample means of random variables that do not have the exponential tilting form. Other examples involving Markov chains and diffusions are presented to illustrate our results.

中文翻译：

---

基于重要性采样的高效大偏差估计

我们提出了一个完整的框架，用于确定基于重要性采样的大偏差概率和速率函数估计器的渐近（或对数）效率。该框架依赖于这样一种思想，即在这种情况下，重要性采样完全以两个随机变量的联合大偏差为特征：定义感兴趣的大偏差概率的可观察量和连接原始过程和的似然因子（或 Radon-Nikodym 导数）。重要性抽样中使用的修改过程。我们用这个框架恢复了关于指数倾斜的渐近效率的已知结果，并获得新的必要和充分条件，使过程的一般变化渐近有效。这使我们能够为不具有指数倾斜形式的随机变量的样本均值构建有效估计量的新示例。介绍了涉及马尔可夫链和扩散的其他示例以说明我们的结果。