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

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

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