On Mean Estimation for Heteroscedastic Random Variables

TL;DR

This paper introduces an adaptive estimator for the mean of heteroscedastic symmetric random variables, achieving near-optimal error bounds without prior knowledge of variances.

Contribution

It proposes a permutation-invariant, fully adaptive estimator that handles unknown, differing variances and provides near-optimal error guarantees under mild regularity conditions.

Findings

Estimator achieves error bounds up to logarithmic factors.

Performance adapts to the unknown variance structure.

Results hold under mild distribution regularity assumptions.

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