# Asymptotically efficient estimation of smooth functionals of covariance operators

### Vladimir Koltchinskii

Georgia Institute of Technology, Atlanta, USA

## Abstract

Let $X$ be a centered Gaussian random variable in a separable Hilbert space $H$ with covariance operator $Σ$. We study the 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 $⟨f(Σ^),B⟩,$ $Σ^:=n_{−1}∑_{j=1}X_{j}⊗X_{j}$ being the sample covariance based on $X_{1},…,X_{n}.$ These bounds show that $⟨f(Σ^),B⟩$ is an asymptotically normal estimator of its expectation $E_{Σ}⟨f(Σ^),B⟩$ (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 $r(Σ)≥n_{1/2}$). When $H$ is a finite-dimensional space of dimension $d=o(n)$, we develop a method of bias reduction and construct an estimator $⟨h(Σ^),B⟩$ 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 asymptotic minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of $⟨h(Σ^),B⟩$ in a semi-parametric sense.

## Cite this article

Vladimir Koltchinskii, Asymptotically efficient estimation of smooth functionals of covariance operators. J. Eur. Math. Soc. 23 (2021), no. 3, pp. 765–843

DOI 10.4171/JEMS/1023