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Efficient Estimation of Smooth Functionals of Covariance Operators Koltchinskii, Vladimir


A problem of efficient estimation of a smooth functional of unknown covariance operator $\Sigma$ of a mean zero Gaussian random variable in a Hilbert space based on a sample of its i.i.d. observations will be discussed. The goal is to find an estimator whose distribution is approximately normal with a minimax optimal variance in a setting when either the dimension of the space, or so called effective rank of the covariance operator are allowed to be large (although much smaller than the sample size). This problem has been recently solved in our joint paper with Loeffler and Nickl in the case of estimation of a linear functional of unknown eigenvector of $\Sigma$ corresponding to its largest eigenvalue (the top principal component). The efficient estimator developed in this paper does not coincide with the naive estimator based on the top principal component of sample covariance which is not efficient due to its large bias. An approach to a more general problem of efficient estimation of a functional $\langle f(\Sigma), B\rangle$ for a given sufficiently smooth function $f:{\mathbb R}\mapsto {\mathbb R}$ and given operator $B$ will be also discussed.

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