Asymptotically efficient estimation of smooth functionals of covariance operators

Abstract

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 $⟨f(\Sigma ),B⟩,$ 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 $⟨f(\hat{\Sigma }),B⟩,$ $\hat{\Sigma }:=n^{-1}{\sum }_{j=1}^n{X}_j\otimes X_j$ being the sample covariance based on $X_1,\dots ,X_n.$ These bounds show that $⟨f(\hat{\Sigma }),B⟩$ is an asymptotically normal estimator of its expectation ${\mathbb{E}}_{\Sigma }⟨f(\hat{\Sigma }),B⟩$ (rather than of parameter of interest $⟨f(\Sigma ),B⟩$) with a parametric convergence rate $O(n^{-1/2})$ provided that the effective rank $\mathbf{r}(\Sigma ):=\mathrm{tr}(\Sigma )\Vert \Sigma \Vert (\mathrm{tr}(\Sigma )$ being the trace and $\Vert \Sigma \Vert$ 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 $\mathbf{r}(\Sigma )\ge n^{1/2}$). When $\mathbb{H}$ is a finite-dimensional space of dimension $d=o(n)$, we develop a method of bias reduction and construct an estimator $⟨h(\hat{\Sigma }),B⟩$ of $⟨f(\Sigma ),B⟩$ that is asymptotically normal with convergence rate $O(n^{-1/2})$. Moreover, we study asymptotic properties of the risk of this estimator and prove asymptotic minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of $⟨h(\hat{\Sigma }),B⟩$ in a semi-parametric sense.