论文标题
量度和随机矩阵方法的集中度
A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics
论文作者
论文摘要
本文研究了数据集合的\ emph {robust协方差矩阵估计} $ x =（x_1，\ ldots，x_n）$，带有$ x_i = \ sqrt ft \ sqrt f i z_i + m $在$ n $和$ p $都大的假设下，$ m \ in \ in \ mathbb r^p $ a \ mathbb r^p $一个确定性信号和$τ_i\ in \ mathbb r $ a的标量扰动可能是较大的振幅。该估计器定义为我们显示的函数的固定点，该函数正在为所谓的\ textit {稳定的半米}收缩。我们利用这一半现象以及量度参数的集中度来证明稳健估计器的存在和独特性，并评估其限制光谱分布。
This article studies the \emph{robust covariance matrix estimation} of a data collection $X = (x_1,\ldots,x_n)$ with $x_i = \sqrt τ_i z_i + m$, where $z_i \in \mathbb R^p$ is a \textit{concentrated vector} (e.g., an elliptical random vector), $m\in \mathbb R^p$ a deterministic signal and $τ_i\in \mathbb R$ a scalar perturbation of possibly large amplitude, under the assumption where both $n$ and $p$ are large. This estimator is defined as the fixed point of a function which we show is contracting for a so-called \textit{stable semi-metric}. We exploit this semi-metric along with concentration of measure arguments to prove the existence and uniqueness of the robust estimator as well as evaluate its limiting spectral distribution.
