Technical Note—Bootstrap-based Budget Allocation for Nested Simulation

Nested simulation (also referred to as two-level simulation) finds a variety of applications such as financial risk measurement, and a central issue of nested simulation is how to allocate a finite amount of simulation budget to achieve the highest accuracy. In “Bootstrap-based Budget Allocation for Nested Simulation”, Zhang, Liu, and Wang propose a bootstrap-based rule for simulation budget allocation for nested simulation. By utilizing the asymptotically optimal inner- and outer-level sample sizes that are typically unknown, the proposed method employs bootstrap sampling on a small amount of initial samples to estimate the unknown optimal sample sizes, thus providing a reasonably good allocation rule for the main simulation. An allocation rule to ensure the asymptotic validity of confidence intervals is also given. Simulation budget allocation is at the heart of a nested (also referred to as two-level) simulation approach to estimating functionals of a conditional expectation. In this paper, we propose a sample-driven budget allocation rule under a unified nested simulation framework that allows for different forms of functionals. The proposed method employs bootstrap sampling to guide an effective choice of outer- and inner-level sample sizes. Furthermore, we establish a central limit theorem for nested simulation estimators, and incorporate the sample-driven allocation rule into the construction of asymptotically valid confidence intervals (CIs). Effectiveness of the sample-driven allocation rule and validity of the constructed CIs are confirmed by numerical experiments.