# Asymptotically Efficient Estimation of Smooth Functionals of Covariance Operators

Let X be a centered Gaussian random variable in a separable Hilbert space H with covariance operator Σ. We study a problem of estimation of a smooth functional of Σ based on a sample X_1,... ,X_n of n independent observations of X. More specifically, we are interested in functionals of the form 〈 f(Σ), B〉, where f: R R is a smooth function and B is a nuclear operator in H. We prove concentration and normal approximation bounds for plug-in estimator 〈 f(Σ̂),B〉, Σ̂:=n^-1∑_j=1^n X_j⊗ X_j being the sample covariance based on X_1,..., X_n. These bounds show that 〈 f(Σ̂),B〉 is an asymptotically normal estimator of its expectation E_Σ〈 f(Σ̂),B〉 (rather than of parameter of interest 〈 f(Σ),B〉) with a parametric convergence rate O(n^-1/2) provided that the effective rank r(Σ):= tr(Σ)/Σ ( tr(Σ) being the trace and Σ being the operator norm of Σ) satisfies the assumption r(Σ)=o(n). At the same time, we show that the bias of this estimator is typically as large as r(Σ)/n (which is larger than n^-1/2 if r(Σ)≥ n^1/2). In the case when H is finite-dimensional space of dimension d=o(n), we develop a method of bias reduction and construct an estimator 〈 h(Σ̂),B〉 of 〈 f(Σ),B〉 that is asymptotically normal with convergence rate O(n^-1/2). Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of 〈 h(Σ̂),B〉 in a semi-parametric sense.

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