On Mean Estimation for Heteroscedastic Random Variables

We study the problem of estimating the common mean $\mu$ 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 a fully adaptive estimator $\widehat{\mu}$ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability, $$|\widehat{\mu} - \mu| \lesssim \min\left\{\sigma_{m^*}, \frac{\sqrt{n}}{\sum_{i = \sqrt{n}}^n \sigma_i^{-1}} \right\}~,$$ where the index $m^* \lesssim \sqrt{n}$ satisfies $m^* \approx \sqrt{\sigma_{m^*}\sum_{i = m^*}^n\sigma_i^{-1}}$.