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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 low rank structure and the smoothness of the covariance function. The new procedure is based on nuclear norm penalization and exhibits superior performances as compared to the sample covariance function by a polynomial factor in the sample size $n$. Other results include a minimax lower bound for estimation of low-rank covariance functions showing that our procedure is optimal as well as a scheme to estimate the unknown noise variance of the Gaussian process.
We study a panel data model with general heterogeneous effects where slopes are allowed to vary across both individuals and over time. The key dimension reduction assumption we employ is that the heterogeneous slopes can be expressed as having a fact
We consider the problem of estimating the covariance matrix of a random signal observed through unknown translations (modeled by cyclic shifts) and corrupted by noise. Solving this problem allows to discover low-rank structures masked by the existenc
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 c
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
Fan et al. [$mathit{Annals}$ $mathit{of}$ $mathit{Statistics}$ $textbf{47}$(6) (2019) 3009-3031] proposed a distributed principal component analysis (PCA) algorithm to significantly reduce the communication cost between multiple servers. In this pape