 Randomized symmetric finite-difference derivative estimator

Given noisy evaluations at $x_0+h{Z}_i$, where the $Z_i$ are independent Rademacher random variables, the estimatoraverages one-sided finite differences from both directions. If $|g^{\prime \prime }|\le M$ and the noise variance is ${\sigma }^2$, its mean squared error is at most $h^2{M}^2/4+{\sigma }^2/(N{h}^2)$.