On the Variance of the variance in control variate Monte Carlo

Control variates are a classical variance-reduction technique for Monte Carlo estimators, yet the behavior of the associated variance estimators is less understood. We analyze the second-order properties of a single control variate Monte Carlo estimator under a simple multifidelity cost model. In the balanced case we obtain an exact finite-sample formula for Var ⁡ [ Var ^ ⁢ ( Q ^ CV ) ] {Var[{Var}({Q}CV)]} in terms of fourth-order moments and show that it decays as 𝒪 ⁢ ( N - 3 ) {O(N⁻³)} in the number of high-fidelity samples N , and hence as 𝒪 ⁢ ( C tot - 3 ) {O(Cₜₒₜ⁻³)} in the total cost. We then derive a compact asymptotic approximation via the multivariate delta method and construct a nonparametric bootstrap analogue. A simulation study on a polynomial benchmark with a low-cost surrogate confirms the predicted scaling and demonstrates that both a fourth-moment plug-in estimator and a bootstrap estimator provide accurate, comparable approximations to this variance-of-variance.