On Mean Estimation for Heteroscedastic Random Variables
This addresses a statistical estimation problem for researchers in robust statistics, but it appears incremental as it builds on existing mean estimation frameworks with heteroscedasticity.
The paper tackles the problem of estimating the common mean of independent symmetric random variables with unknown and varying standard deviations, showing that a fully adaptive estimator achieves an error bound that scales with the minimum of two terms involving the standard deviations, up to logarithmic factors.
We study the problem of estimating the common mean $μ$ of $n$ independent symmetric random variables with different and unknown standard deviations $σ_1 \le σ_2 \le \cdots \leσ_n$. We show that, under some mild regularity assumptions on the distribution, there is a fully adaptive estimator $\widehatμ$ 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 $m^* \lesssim \sqrt{n}$ satisfies $m^* \approx \sqrt{σ_{m^*}\sum_{i = m^*}^nσ_i^{-1}}$.