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

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