Estimation of smooth functionals in high-dimensional models: bootstrap chains and Gaussian approximation

Let $X^{(n)}$ be an observation sampled from a distribution $P_{theta}^{(n)}$ with an unknown parameter $theta,$ $theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(theta)$ for a functional $f:Emapsto {mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}sim P_{theta}^{(n)}.$ Assuming that there exists an estimator $hat theta_n=hat theta_n(X^{(n)})$ of parameter $theta$ such that $sqrt{n}(hat theta_n-theta)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:Emapsto {mathbb R}$ such that $g(hat theta_n)$ is an asymptotically normal estimator of $f(theta)$ with $sqrt{n}$ rate provided that $s>frac{1}{1-alpha}$ and $dleq n^{alpha}$ for some $alphain (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(hat theta)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $sqrt{n}(hat theta_n-theta).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.