Nested Simulation Methods for Sobol' Index Estimation: Bias Correction, Budget Allocation, and Latin Hypercube Sampling

2026-07-07

AI Summary

This study addresses bias and efficiency issues in estimating Sobol’ indices for global sensitivity analysis by unifying classical pick-freeze and nested Monte Carlo estimators within a nested simulation framework, revealing that the former is a special case with fixed inner-sample size. Building on this insight, the authors propose two novel jackknife-based estimators—unbiased jackknife and split-sample jackknife—and provide the first systematic evaluation of how Latin hypercube sampling (LHS) affects bias correction across different estimators. Theoretical analysis demonstrates that the split-sample jackknife estimator achieves the standard mean squared error (MSE) convergence rate under a fixed computational budget, outperforming conventional approaches. Numerical experiments corroborate the theoretical findings and offer practical guidance for selecting estimators in real-world applications.

Abstract

Estimating the variance of a conditional expectation is a recurring problem in stochastic simulation, with applications in global sensitivity analysis and Sobol' index estimation. This paper revisits Sobol' index estimation through the lens of nested simulation and develops a unified comparison of classical pick-freeze estimators and nested simulation estimators under a common computational budget. We show that several standard pick-freeze estimators can be interpreted as nested simulation estimators with fixed inner-level sample sizes, enabling direct performance comparisons and clarifying their bias-variance behavior. Building on this perspective, we analyze the standard nested simulation estimator for the Sobol' index numerator and propose two jackknife-based extensions: an unbiased jackknife estimator and a split jackknife estimator that uses an independent preliminary sample to estimate the mean. Under crude Monte Carlo (CMC), the split jackknife estimator attains the canonical mean squared error (MSE) rate, whereas the standard nested simulation and unbiased jackknife estimators attain the slower nested simulation rate. We also characterize the associated allocations of outer- and inner-level simulation effort. Finally, we study the impact of Latin hypercube sampling (LHS), showing that it can improve the standard nested simulation estimator while undermining bias reduction in jackknife-based estimators unless the inner-level sample size grows with the total budget. Numerical experiments corroborate the theory and provide practical guidance on estimator selection for Sobol' index estimation under CMC and LHS.

Problem

• Sobol' index
• nested simulation
• variance estimation
• bias correction
• computational budget

Innovation

• nested simulation
• Sobol' index
• jackknife estimator
• Latin hypercube sampling
• bias correction