Adaptive particle techniques and rare event estimation
Adaptive particle techniques and rare event estimation
复制标题
自适应粒子技术和罕见事件估计
作者：
Cérou F
在可靠性、通信、飞机管理等领域，罕见事件概率的估计是一个至关重要的问题。在复杂系统中，分析研究是不可能的，人们必须使用蒙特卡罗方法。当稀有真的很罕见时，这意味着概率小于10-9，天真的蒙特卡罗就变得不合理了。一种广泛使用的技术是多级拆分，但这种方法需要对系统有足够的了解，才能决定将级别放在哪里。不幸的是，这并不总是可能的。在本文中，我们提出了一种自适应算法来解决这个问题：估计是渐近一致的，代价仅比经典的多水平分裂略高，并且在渐近方差方面具有相同的效率。在一维情形中，我们严格地证明了A.S.我们的估计量的收敛和渐近正态，与其他使用固定交叉水平的算法具有相同的方差。在我们的证明中，我们主要使用经验过程理论中的工具，这在罕见事件领域似乎是相当新的。
The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10-9, naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but this method requires enough knowledge about the system to decide where to put the levels at hand. This is unfortunately not always possible. In this paper, we propose an adaptive algorithm to cope with this problem: the estimation is asymptotically consistent, costs just a little bit more than classical multilevel splitting and has the same efficiency in terms of asymptotic variance. In the one dimensional case, we prove rigorously the a.s. convergence and the asymptotic normality of our estimator, with the same variance as with other algorithms that use fixed crossing levels. In our proofs we mainly use tools from the theory of empirical processes, which seems to be quite new in the field of rare events.