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

Abstract

Let X(n) be an observation sampled from a distribution Pθ(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↦R of some smoothness s>0 based on an observation X(n)∼Pθ(n). Assuming that there exists an estimator θˆn=θˆn(X(n)) of parameter θ such that n(θˆ n−θ) is sufficiently close in distribution to a mean zero Gaussian random vector in E, we construct a functional g:E↦R such that g(θˆn) is an asymptotically normal estimator of f(θ) with n rate provided that s>11−α and d≤nα for some α∈(0,1). We also derive general upper bounds on Orlicz norm error rates for estimator g(θˆ) depending on smoothness s, dimension d, sample size n and the accuracy of normal approximation of n(θˆ n−θ). In particular, this approach yields asymptotically efficient estimators in high-dimensional log-concave exponential models.