Many two-level nested simulation applications involve the conditional expectation of some response variable, where the expected response is the quantity of interest, and the expectation is with respect to the inner-level random variables, conditioned on the outer-level random variables. The latter typically represent random risk factors, and risk can be quantified by estimating the probability density function (pdf) or cumulative distribution function (cdf) of the conditional expectation. Much prior work has considered a naïve estimator that uses the empirical distribution of the sample averages across the inner-level replicates. This results in a biased estimator, because the distribution of the sample averages is over-dispersed relative to the distribution of the conditional expectation when the number of inner-level replicates is finite. Whereas most prior work has focused on allocating the numbers of outer- and inner-level replicates to balance the bias/variance tradeoff, we develop a bias-corrected pdf estimator. Our approach is based on the concept of density deconvolution, which is widely used to estimate densities with noisy observations but has not previously been considered for nested simulation problems. For a fixed computational budget, the bias-corrected deconvolution estimator allows more outer-level and fewer inner-level replicates to be used, which substantially improves the efficiency of the nested simulation.

中文翻译：

---

嵌套模拟问题中条件期望密度的偏差校正估计

许多两级嵌套模拟应用程序涉及某些响应变量的条件期望，其中期望响应是感兴趣的数量，期望是关于内部随机变量的，以外部随机变量为条件。后者通常代表随机风险因素，可以通过估计条件期望的概率密度函数（pdf）或累积分布函数（cdf）来量化风险。许多先前的工作都考虑了一个简单的估计器，它使用样本平均值在内部级别复制中的经验分布。这会导致估计量有偏差，因为当内部级别重复的数量有限时，样本平均值的分布相对于条件期望的分布是过度分散的。尽管大多数先前的工作都集中在分配外层和内层复制的数量以平衡偏差/方差权衡，但我们开发了一个偏差校正的 pdf 估计器。我们的方法基于密度反卷积的概念，该概念被广泛用于估计具有噪声观测的密度，但以前并未考虑用于嵌套模拟问题。对于固定的计算预算，偏差校正反卷积估计器允许使用更多的外层和更少的内层复制，这大大提高了嵌套模拟的效率。它广泛用于估计具有噪声观察的密度，但以前没有考虑过用于嵌套模拟问题。对于固定的计算预算，偏差校正反卷积估计器允许使用更多的外层和更少的内层复制，这大大提高了嵌套模拟的效率。它广泛用于估计具有噪声观察的密度，但以前没有考虑过用于嵌套模拟问题。对于固定的计算预算，偏差校正反卷积估计器允许使用更多的外层和更少的内层复制，这大大提高了嵌套模拟的效率。