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We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_1,dots, X_n$ in a separable Hilbert space $mathbb{H}$ with unknown covariance operator $Sigma.$ The complexity of the problem is characterized by its effective rank ${bf r}(Sigma):= frac{{rm tr}(Sigma)}{|Sigma|},$ where ${rm tr}(Sigma)$ denotes the trace of $Sigma$ and $|Sigma|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $Sigma.$ Under the assumption that ${bf r}(Sigma)=o(n),$ we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semi-parametric optimality.
Principal component analysis is an important pattern recognition and dimensionality reduction tool in many applications. Principal components are computed as eigenvectors of a maximum likelihood covariance $widehat{Sigma}$ that approximates a populat
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
We study a problem of estimation of smooth functionals of parameter $theta $ of Gaussian shift model $$ X=theta +xi, theta in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $theta$ in Gaussian noise $xi$ with
In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. Especially, we derive th
Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alte