Asymptotically efficient estimation of smooth functionals of covariance operators

Let X be a centered Gaussian random variable in a separable Hilbert space {\mathbb H} with covariance operator \Sigma . We study the problem of estimation of a smooth functional of \Sigma based on a sample X_1,\dots ,X_n of n independent observations of X . More specifically, we are interested in functionals of the form \langle f(\Sigma), B\rangle, where f:{\mathbb R}\mapsto {\mathbb R} is a smooth function and B is a nuclear operator in {\mathbb H} . We prove concentration and normal approximation bounds for plug-in estimator \langle f(\hat \Sigma),B\rangle, \hat \Sigma:=n^{-1}\sum_{j=1}^n X_j\otimes X_j being the sample covariance based on X_1,\dots, X_n. These bounds show that \langle f(\hat \Sigma),B\rangle is an asymptotically normal estimator of its expectation {\mathbb E}_{\Sigma} \langle f(\hat \Sigma),B\rangle (rather than of parameter of interest \langle f(\Sigma),B\rangle ) with a parametric convergence rate O(n^{-1/2}) provided that the effective rank {\bf r}(\Sigma):= \mathrm {tr}(\Sigma)\|\Sigma\| (\mathrm {tr} (\Sigma) being the trace and \|\Sigma\| being the operator norm of \Sigma ) satisfies the assumption {\mathbf r}(\Sigma)=o(n). At the same time, we show that the bias of this estimator is typically as large as \mathbf r(\Sigma)/{n} (which is larger than n^{-1/2} if \mathbf r(\Sigma)\geq n^{1/2} ). When \mathbb H is a finite-dimensional space of dimension d=o(n) , we develop a method of bias reduction and construct an estimator \langle h(\hat \Sigma),B\rangle of \langle f(\Sigma),B\rangle that is asymptotically normal with convergence rate O(n^{-1/2}) . Moreover, we study asymptotic properties of the risk of this estimator and prove asymptotic minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of \langle h(\hat \Sigma),B\rangle in a semi-parametric sense.