Abstract
Let be an observation sampled from a distribution 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 for a functional of some smoothness based on an observation . Assuming that there exists an estimator of parameter θ such that is sufficiently close in distribution to a mean zero Gaussian random vector in E, we construct a functional such that is an asymptotically normal estimator of with rate provided that and for some . We also derive general upper bounds on Orlicz norm error rates for estimator depending on smoothness s, dimension d, sample size n and the accuracy of normal approximation of . In particular, this approach yields asymptotically efficient estimators in high-dimensional log-concave exponential models.
Funding Statement
Supported in part by NSF Grants DMS-1810958 and DMS-2113121.
Acknowledgment
The author is very thankful to Clément Deslandes for careful reading of the manuscript and suggesting a number of corrections, and to the referees for helpful comments.
Citation
Vladimir Koltchinskii. "Estimation of smooth functionals in high-dimensional models: Bootstrap chains and Gaussian approximation." Ann. Statist. 50 (4) 2386 - 2415, August 2022. https://doi.org/10.1214/22-AOS2197
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