Estimation of smooth functionals of covariance operators: jackknife bias reduction and bounds in terms of effective rank

TL;DR

This paper investigates the estimation of smooth functionals of covariance operators in infinite-dimensional Banach spaces, establishing bias reduction techniques and bounds based on effective rank, with conditions for achieving classical convergence rates and asymptotic normality.

Contribution

It extends covariance functional estimation methods to infinite-dimensional Banach spaces, analyzing bias reduction and error bounds in terms of effective rank and smoothness.

Findings

Classical $\,\sqrt{n}$-rate achievable under certain effective rank and smoothness conditions.

Asymptotic normality and efficiency established for higher smoothness levels.

Results generalize previous finite-dimensional covariance estimation to infinite-dimensional settings.

Abstract

Let $E$ be a separable Banach space and let $X, X_1,\dots, X_n, \dots$ be i.i.d. Gaussian random variables taking values in $E$ with mean zero and unknown covariance operator $\Sigma: E^{\ast}\mapsto E.$ The complexity of estimation of $\Sigma$ based on observations $X_1,\dots, X_n$ is naturally characterized by the so called effective rank of $\Sigma:$ ${\bf r}(\Sigma):= \frac{{\mathbb E}_{\Sigma}\|X\|^2}{\|\Sigma\|},$ where $\|\Sigma\|$ is the operator norm of $\Sigma.$ Given a smooth real valued functional $f$ defined on the space $L(E^{\ast},E)$ of symmetric linear operators from $E^{\ast}$ into $E$ (equipped with the operator norm), our goal is to study the problem of estimation of $f(\Sigma)$ based on $X_1,\dots, X_n.$ The estimators of $f(\Sigma)$ based on jackknife type bias reduction are considered and the dependence of their Orlicz norm error rates on effective rank ${\bf r}(\Sigma),$ the sample size $n$ and the degree of H\"older smoothness $s$ of functional $f$ are studied. In particular, it is shown that, if ${\bf r}(\Sigma)\lesssim n^{\alpha}$ for some $\alpha\in (0,1)$ and $s\geq \frac{1}{1-\alpha},$ then the classical $\sqrt{n}$-rate is attainable and, if $s> \frac{1}{1-\alpha},$ then asymptotic normality and asymptotic efficiency of the resulting estimators hold. Previously, the results of this type (for different estimators) were obtained only in the case of finite dimensional Euclidean space $E={\mathbb R}^d$ and for covariance operators $\Sigma$ whose spectrum is bounded away from zero (in which case, ${\bf r}(\Sigma)\asymp d$).