论文标题
catoni的$ {\ rm m} $ - 有限{$α$ - themment ashaption}的估计器，$α\ in（1,2）$
A generalized Catoni's ${\rm M}$-estimator under finite {$α$-th moment assumption} with $α\in (1,2)$
论文作者
论文摘要
我们将Catoni在他的开创性论文[C12]中推出的{$ {\ rm m} $ - 估算器}将样本可以具有有限的$α$ -th时刻（1,2）$而不是有限差异的情况，我们的方法是通过稍微修改影响函数$φ$φ$φ$。新影响函数的选择灵感来自[C-N-X]中开发的类似泰勒的膨胀。我们获得了估计器的偏差，为$α\ rightarrow 2 $，该界限与[C12]中的界限相同。实验表明，我们的广义$ {\ rm m} $ - 估算器的性能要比经验平均值估计器更好，$α$的越小，性能就会越好。作为一个应用程序，我们研究了Zhang等人考虑的$ \ ell_ {1} $回归。 [Z-Z]假设样品具有有限的差异，并放宽其假设为有限的{$α$ -th}时刻，$α\ in（1,2）$。
We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminal paper [C12] to the case in which samples can have finite $α$-th moment with $α\in (1,2)$ rather than finite variance, our approach is by slightly modifying the influence function $φ$ therein. The choice of the new influence function is inspired by the Taylor-like expansion developed in [C-N-X]. We obtain a deviation bound of the estimator, as $α\rightarrow 2$, this bound is the same as that in [C12]. Experiment shows that our generalized ${\rm M}$-estimator performs better than the empirical mean estimator, the smaller the $α$ is, the better the performance will be. As an application, we study an $\ell_{1}$ regression considered by Zhang et al. [Z-Z] who assumed that samples have finite variance, and relax their assumption to be finite {$α$-th} moment with $α\in (1,2)$.
