论文标题
$ \ mathbb {r} $中的最佳亚高斯平均估计
Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$
论文作者
论文摘要
我们重新审视了估计实数分布的平均值的问题，它以次高斯融合的新估计量：直觉上，“我们的估计器在任何分布上，与样本平均值一样准确，是用于匹配方差的高斯分布。”至关重要的是，与先前的工作相反，我们的估计器不需要对差异的先验知识，并且在整个分布范围内都具有有界差异的范围，包括没有任何较高时刻的那些。由样本大小$ n $，失败概率$δ$和方差$σ^2 $进行参数，我们的估计器准确地在$σ\ cdot（1+o（1+o（1））\ sqrt {\ frac {\ frac {2 \ log \ log \ frac \ frac {1}}}Δ}} {n} {n} {n}} $（$ 1）$（$ 1）$（1+o（1）$ 1+o（我们的估算构建和分析提供了一个可以推广到其他问题的框架，通过将总和隐式视为2参数$ψ$估计器，从而严格地分析了相关的随机变量总和，并使用数学编程和二元技术来构建界限。
We revisit the problem of estimating the mean of a real-valued distribution, presenting a novel estimator with sub-Gaussian convergence: intuitively, "our estimator, on any distribution, is as accurate as the sample mean is for the Gaussian distribution of matching variance." Crucially, in contrast to prior works, our estimator does not require prior knowledge of the variance, and works across the entire gamut of distributions with bounded variance, including those without any higher moments. Parameterized by the sample size $n$, the failure probability $δ$, and the variance $σ^2$, our estimator is accurate to within $σ\cdot(1+o(1))\sqrt{\frac{2\log\frac{1}δ}{n}}$, tight up to the $1+o(1)$ factor. Our estimator construction and analysis gives a framework generalizable to other problems, tightly analyzing a sum of dependent random variables by viewing the sum implicitly as a 2-parameter $ψ$-estimator, and constructing bounds using mathematical programming and duality techniques.
