  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 $m^* \approx \sqrt{\sigma_{m^*}\sum_{i
= m^*}^n\sigma_i^{-1}}$.
