On Mean Estimation for Heteroscedastic Random Variables

We study the problem of estimating the common mean μ of n independent symmetric random variables with different and unknown standard deviations σ1 ≤ σ2 ≤ cdots ≤σn. We show that, under some mild regularity assumptions on the distribution, there is a fully adaptive estimator widehatμ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability, [ |widehatμ - μ| lesssim minleftσm^*, frac√nsum_i = √nn σi-1 right~, ] where the index m^* lesssim √n satisfies m^* ≈ √σm^*sumi = m^*nσi-1.