论文标题
关于异质的随机变量的平均估计
On Mean Estimation for Heteroscedastic Random Variables
论文作者
论文摘要
我们研究了估计具有不同和未知标准偏差的$ n $独立对称的随机变量$σ_1\leσ_2\ le \ cdots \ cdots \leσ_n$的问题的问题。我们表明，在分布的一些轻度规律性假设下，有一个完全自适应的估计量$ \wideHatμ$，因此它不变到样本元素的排列，并满足以对数因素的高概率，并且具有很高的可能性，\ [| \wideHatμ -μ -μ| | \ sillssim \ min \ left \ {σ_{m^*}，\ frac {\ sqrt {n}}} {\ sum _ {\ sum _ {i = \ sqrt {n}}}}}^nσ_i^{ - 1}}}}}}}}}}}}}} \ right \} \ sqrt {n} $满足$ m^*\ oit sqrt {σ_{m^*} \ sum_ {i = m^*}^nσ_i^{ - 1}}} $。
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 $m^* \approx \sqrt{σ_{m^*}\sum_{i = m^*}^nσ_i^{-1}}$.
