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 $\mathrm{\Sigma }\mathrm{.}$ We study a problem of estimation of a smooth functional of $\mathrm{\Sigma }$ based on a sample $X_1,\dots ,X_n$of$n$independentobservationsof$X.$ More specifically, we are interested in functionals of the form $⟨f(\mathrm{\Sigma }),B⟩,$ where $f:\mathbb{R}\mapsto \mathbb{R}$ is a smooth function and $B$ is a nuclear operator in $\mathbb{H}\mathrm{.}$ We prove concentration and normal approximation bounds for plug-in estimator $⟨f(\widehat{\mathrm{\Sigma }}),B⟩,$ $\hat \Sigma:=n^{-1}\sum_{j=1}^n X_j\otimes X_j$beingthesamplecovariancebasedon$X_1,\dots, X_n.$Theseboundsshowthat$\langle f(\hat \Sigma),B\rangle$ is an asymptotically normal estimator of its expectation ${\mathbb E}_{\Sigma} \langle f(\hat \Sigma),B\rangle$(ratherthanofparameterofinterest$\langle f(\Sigma),B\rangle$)withaparametricconvergencerate$O(n^{-1/2})$ provided that the effective rank $\mathbf{r}(\mathrm{\Sigma }):=\frac{\mathbf{t}\mathbf{r}(\mathrm{\Sigma })}{\mathrm{\parallel }\mathrm{\Sigma }\mathrm{\parallel }}$ ($\mathrm{t}\mathrm{r}(\mathrm{\Sigma })$ being the trace and $\mathrm{\parallel }\mathrm{\Sigma }\mathrm{\parallel }$ being the operator norm of $\mathrm{\Sigma }$) satisfies the assumption $\mathbf{r}(\mathrm{\Sigma })=o(n)\mathrm{.}$ At the same time, we show that the bias of this estimator is typically as large as $\frac{{\bf r}(\Sigma)}{n}$(whichislargerthan$n^{-1/2}$if${\bf r}(\Sigma)\geq n^{1/2}$)\mathrm{.}Inthecasewhen${\mathbb H}$ is finite-dimensional space of dimension $d=o(n),$ we develop a method of bias reduction and construct an estimator $⟨h(\widehat{\mathrm{\Sigma }}),B⟩$ of $⟨f(\mathrm{\Sigma }),B⟩$ that is asymptotically normal with convergence rate $O(n^{-1\mathrm{/}2})\mathrm{.}$ Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of $⟨h(\widehat{\mathrm{\Sigma }}),B⟩$ in a semi-parametric sense.