# On Mean Estimation for Heteroscedastic Random Variables

arXiv: https://arxiv.org/abs/2010.11537 · HTML: https://mlsanity.com/paper/2010.11537/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2020.

## Built on by (most cited first, 5 of 5)

- [Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination](https://mlsanity.com/paper/2502.14772/) (2025)
- [Near-Optimal Mean Estimation with Unknown, Heteroskedastic Variances](https://mlsanity.com/paper/2312.02417/) (2023)
- [Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm](https://mlsanity.com/paper/2411.15660/) (2024)
- [Entangled Mean Estimation in High-Dimensions](https://mlsanity.com/paper/2501.05425/) (2025)
- [Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation](https://mlsanity.com/paper/2502.05730/) (2025)

## Abstract (arXiv)

We study the problem of estimating the common mean μ of n independent symmetric random variables with different and unknown standard deviations σ1 ≤ σ2 ≤ cdots ≤σn. 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 minleftσm^*, frac√nsum_i = √nn σi-1 right~, ] where the index m^* lesssim √n satisfies m^* ≈ √σm^*sumi = m^*nσi-1.