Finite-Sample Symmetric Mean Estimation with Fisher Information Rate

The mean of an unknown variance-σ2 distribution f can be estimated from n samples with variance fracσ2n and nearly corresponding subgaussian rate. When f is known up to translation, this can be improved asymptotically to frac1nmathcal I, where mathcal I is the Fisher information of the distribution. Such an improvement is not possible for general unknown f, but [Stone, 1975] showed that this asymptotic convergence textitis possible if f is textitsymmetric about its mean. Stone's bound is asymptotic, however: the n required for convergence depends in an unspecified way on the distribution f and failure probability δ. In this paper we give finite-sample guarantees for symmetric mean estimation in terms of Fisher information. For every f, n, δ with n > log frac1δ, we get convergence close to a subgaussian with variance frac1n mathcal Ir, where mathcal Ir is the r-textitsmoothed Fisher information with smoothing radius r that decays polynomially in n. Such a bound essentially matches the finite-sample guarantees in the known-f setting.