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We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distribution, such as existence of moments of only low order. While estimation of covariance matrices corresponding to sub-Gaussian distributions is well-understood, much less in known in the case of heavy-tailed data. As K. Balasubramanian and M. Yuan write, data from real-world experiments oftentimes tend to be corrupted with outliers and/or exhibit heavy tails. In such cases, it is not clear that those covariance matrix estimators .. remain optimal and ..what are the other possible strategies to deal with heavy tailed distributions warrant further studies. We make a step towards answering this question and prove tight deviation inequalities for the proposed estimator that depend only on the parameters controlling the intrinsic dimension associated to the covariance matrix (as opposed to the dimension of the ambient space); in particular, our results are applicable in the case of high-dimensional observations.
We offer a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce element-
We consider high-dimensional multivariate linear regression models, where the joint distribution of covariates and response variables is a multivariate normal distribution with a bandable covariance matrix. The main goal of this paper is to estimate
We consider the fitting of heavy tailed data and distribution with a special attention to distributions with a non--standard shape in the body of the distribution. To this end we consider a dense class of heavy tailed distributions introduced recentl
We consider the problem of estimating a low rank covariance function $K(t,u)$ of a Gaussian process $S(t), tin [0,1]$ based on $n$ i.i.d. copies of $S$ observed in a white noise. We suggest a new estimation procedure adapting simultaneously to the lo
Let $X$ be a centered Gaussian random variable in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma.$ We study a problem of estimation of a smooth functional of $Sigma$ based on a sample $X_1,dots ,X_n$ of $n$ independent observa