Technical note—Constructing confidence intervals for nested simulation

Abstract

Nested simulation is typically used to estimate the functional of a conditional expectation. Considerable research has been performed on point estimation for various functionals. However, the quantification of the statistical uncertainty in the point estimator, for instance, using confidence intervals (CIs), has not been extensively investigated. In this article, we establish central limit theorems with the asymptotically optimal convergence rate of Γ−1false/3$$ {\Gamma}^{-1/3} $$ for nested simulation with different forms of functionals, where normalΓ$$ \Gamma $$ denotes the total computational effort. Based on these theorems, we develop a unified CI framework that can ensure that both the mean squared error of the point estimator and CI width attain the optimal convergence rate. Numerical examples are presented, and the results are found to be consistent with the theoretical results. Experimental results demonstrate that the proposed framework outperforms the existing methods for CI construction in terms of the CI widths and convergence rates.