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Near-Optimal Mean Estimation with Unknown, Heteroskedastic Variances

arXiv - MATH - Statistics Theory Pub Date : 2023-12-05 DOI:arxiv-2312.02417
Spencer Compton, Gregory Valiant
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Abstract

Given data drawn from a collection of Gaussian variables with a common mean but different and unknown variances, what is the best algorithm for estimating their common mean? We present an intuitive and efficient algorithm for this task. As different closed-form guarantees can be hard to compare, the Subset-of-Signals model serves as a benchmark for heteroskedastic mean estimation: given $n$ Gaussian variables with an unknown subset of $m$ variables having variance bounded by 1, what is the optimal estimation error as a function of $n$ and $m$? Our algorithm resolves this open question up to logarithmic factors, improving upon the previous best known estimation error by polynomial factors when $m = n^c$ for all $0
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具有未知异方差的近最优均值估计
给定从高斯变量集合中提取的数据，这些变量具有共同的平均值，但方差不同且未知，那么估计它们的共同平均值的最佳算法是什么?我们提出了一种直观有效的算法。由于不同的封闭形式保证很难比较，信号子集模型作为异方差均值估计的基准:给定$n$高斯变量与未知的$m$变量子集，其方差以1为界，作为$n$和$m$的函数，最优估计误差是什么?我们的算法通过拓扑因素解决了这个开放的问题，当$m = n^c$对于所有$0
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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arXiv - MATH - Statistics Theory
arXiv - MATH - Statistics Theory
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