Estimating nested expectations is an important task in computational mathematics and statistics. In this paper we propose a new Monte Carlo method using post-stratification to estimate nested expectations efficiently without taking samples of the inner random variable from the conditional distribution given the outer random variable. This property provides the advantage over many existing methods that it enables us to estimate nested expectations only with a dataset on the pair of the inner and outer variables drawn from the joint distribution. We show an upper bound on the mean squared error of the proposed method under some assumptions. Numerical experiments are conducted to compare our proposed method with several existing methods (nested Monte Carlo method, multilevel Monte Carlo method, and regression-based method), and we see that our proposed method is superior to the compared methods in terms of efficiency and applicability.
翻译:估计嵌套期望是计算数学和统计方面的一项重要任务。 在本文中,我们提议采用一个新的蒙特卡洛方法,使用批准后的方法,有效估计嵌套期望,而不必从外部随机变量的有条件分布中抽取内部随机变量样本。这一属性比许多现有方法的优势在于,它只能用从联合分布中提取的内外部变量的数据集来估计嵌套期望。我们在某些假设中显示了拟议方法的平均正方形错误的上限。我们进行了数字实验,将我们提议的方法与几种现有方法(取消的蒙特卡洛方法、多层次的蒙特卡洛方法和回归法)进行比较,我们看到,我们提议的方法在效率和适用性方面优于比较方法。