Nearly optimal robust mean estimation via empirical characteristic function

We propose an estimator for the mean of random variables in separable real Banach spaces using empirical characteristic function. Assuming that covariance operator variable is bounded a precise sense, we show proposed achieves optimal sub-Gaussian rate, except faster decaying mean-dependent term. Under mild condition, iterative refinement procedure can essentially eliminate term and provide shift-equivariant estimate. Our results particularly suggests certain Gaussian width appears best known rate literature might not be necessary. Furthermore, boundedness functions, also that, possibly at high signal-to-noise ratios, gracefully robust against adversarial “contamination”. analysis overall concise transparent, thanks to tractability functions.