The goal of this paper is to show that a single robust estimator of the mean
of a multivariate Gaussian distribution can enjoy five desirable properties.
First, it is computationally tractable in the sense that it can be computed in
a time which is at most polynomial in dimension, sample size and the logarithm
of the inverse of the contamination rate. Second, it is equivariant by
translations, uniform scaling and orthogonal transformations. Third, it has a
high breakdown point equal to $0.5$, and a nearly-minimax-rate-breakdown point
approximately equal to $0.28$. Fourth, it is minimax rate optimal, up to a
logarithmic factor, when data consists of independent observations corrupted by
adversarially chosen outliers. Fifth, it is asymptotically efficient when the
rate of contamination tends to zero. The estimator is obtained by an iterative
reweighting approach. Each sample point is assigned a weight that is
iteratively updated by solving a convex optimization problem. We also establish
a dimension-free non-asymptotic risk bound for the expected error of the
proposed estimator. It is the first result of this kind in the literature and
involves only the effective rank of the covariance matrix. Finally, we show
that the obtained results can be extended to sub-Gaussian distributions, as
well as to the cases of unknown rate of contamination or unknown covariance
matrix.

本文研究了一个基于迭代重新加权的估计方法，该方法针对多元高斯分布的均值具有鲁棒性，且具有多个优秀性质，包括计算上的可行性、对平移、伸缩和正交变换的不变性、高断点以及渐近有效性。此外，本文还为提出的估计器建立了无维度的非渐近风险界限，并将结果推广到了子高斯分布和污染率未知、协方差矩阵未知等情形。

高斯均值全方位健壮估计器

All-In-One Robust Estimator of the Gaussian Mean

We compare two recently proposed methods that combine ideas from conformal
inference and quantile regression to produce locally adaptive and marginally
valid prediction intervals under sample exchangeability (Romano et al., 2019;
Kivaranovic et al., 2019). First, we prove that these two approaches are
asymptotically efficient in large samples, under some additional assumptions.
Then we compare them empirically on simulated and real data. Our results
demonstrate that the method in Romano et al. (2019) typically yields tighter
prediction intervals in finite samples. Finally, we discuss how to tune these
procedures by fixing the relative proportions of observations used for training
and conformalization.

本文通过比较两种新提出的置信推断和分位数回归思想相结合的方法，并考虑在样本交换性下如何产生局部自适应和边缘有效的预测区间，证明当一些额外的假定满足时，这两种方法在大样本下是渐进有效的，同时在模拟和真实数据中进行实证比较，结果表明相对于 Kivaranovic 等人 (2019) 的方法，Romano 等人 (2019) 的方法在有限样本中通常产生更紧的预测区间，并最后讨论了如何通过调整用于训练和适应性的相对比例来调整这些程序。