On Mean Estimation for Heteroscedastic Random Variables

Abstract

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 $\hat{\mu }$ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability, \[ |\widehatμ - μ| \lesssim \min\left\{σ_{m^*}, \frac{\sqrt{n}}{\sum_{i = \sqrt{n}}^n σ_i^{-1}} \right\}~, \] where the index $m^{\ast }\lesssim \sqrt{n}$ satisfies $m^{\ast }\approx \sqrt{{\sigma }_{m^{\ast }}{\sum }_{i=m^{\ast }}^n{\sigma }_i^{-1}}$.

29 pages