Efficient Estimation of Smooth Functionals in Gaussian Shift Models

Abstract

We study a problem of estimation of smooth functionals of parameter $\theta $ of Gaussian shift model $$ X=\theta +\xi,\ \theta \in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $\theta$ in Gaussian noise $\xi$ with zero mean and known covariance operator $\Sigma.$ In particular, we develop estimators $T(X)$ of $f(\theta)$ for functionals $f:E\mapsto {\mathbb R}$ of H\"older smoothness $s>0$ such that $$ \sup_{\|\theta\|\leq 1} {\mathbb E}_{\theta}(T(X)-f(\theta))^2 \lesssim \Bigl(\|\Sigma\| \vee ({\mathbb E}\|\xi\|^2)^s\Bigr)\wedge 1, $$ where $\|\Sigma\|$ is the operator norm of $\Sigma,$ and show that this mean squared error rate is minimax optimal at least in the case of standard Gaussian shift model ($E={\mathbb R}^d$ equipped with the canonical Euclidean norm, $\xi =\sigma Z,$ $Z\sim {\mathcal N}(0;I_d)$). Moreover, we determine a sharp threshold on the smoothness $s$ of functional $f$ such that, for all $s$ above the threshold, $f(\theta)$ can be estimated efficiently with a mean squared error rate of the order $\|\Sigma\|$ in a "small noise" setting (that is, when ${\mathbb E}\|\xi\|^2$ is small). The construction of efficient estimators is crucially based on a "bootstrap chain" method of bias reduction. The results could be applied to a variety of special high-dimensional and infinite-dimensional Gaussian models (for vector, matrix and functional data).