Estimation of smooth functionals in high-dimensional models: bootstrap chains and Gaussian approximation
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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↦ℝ of some smoothness s>0 based on an observation X^(n)∼ P_θ^(n). Assuming that there exists an estimator θ̂_n=θ̂_n(X^(n)) of parameter θ such that √(n)(θ̂_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(θ̂_n) is an asymptotically normal estimator of f(θ) with √(n) rate provided that s>1/1-α and d≤ n^α for some α∈ (0,1). We also derive general upper bounds on Orlicz norm error rates for estimator g(θ̂) depending on smoothness s, dimension d, sample size n and the accuracy of normal approximation of √(n)(θ̂_n-θ). In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.
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