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 ${\sigma }_1\le {\sigma }_2\le \cdots \le {\sigma }_n$. We show that, under some mild regularity assumptions on the distribution, there is an adaptive estimator $\hat{\mu }$ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability,