Multivariate control variates (${\beta }_{\ast }={\Sigma }^{-1}\mathrm{Cov}(C,Y)$)

If $Y$ is an unbiased estimator and a vector $C$ has known mean, then $Y-{\beta }^T(C-\mathbb{E}C)$ remains unbiased for a deterministic coefficient vector. Its variance is $\mathrm{Var}Y-2{\beta }^{T}c+{\beta }^T\Sigma \beta$, where $c=\mathrm{Cov}(C,Y)$ and $\Sigma =\mathrm{Cov}(C)$. For invertible $\Sigma$, completing the square gives the displayed optimum and minimum variance $\mathrm{Var}Y-c^T{\Sigma }^{-1}c$. With singular $\Sigma$, use a generalized inverse on its range. Estimating coefficients from the same simulation can introduce finite-sample bias; fixed pilot coefficients or independent training preserve the elementary unbiasedness assertion.