Order-Explicit Linearization of High-Dimensional -Statistics
arXiv:2405.07860
The paper derives explicit large‑deviation bounds for high‑dimensional U‑statistics, showing that their deviation from the Hájek projection scales as O_p(ϕ b n⁻¹ log²(dn)) and applies these results to concentration, Gaussian approximation, and resampling‑based confidence intervals for nonparametric regression methods such as random forests.
Abstract
We give an order-explicit large deviation bound for the difference between a high-dimensional -statistic and its Hájek projection. In particular, we show that any -statistic of order on observations, with a -dimensional kernel whose coordinates have -Orlicz norm at most , has a maximum deviation from its Hájek projection of order . The proof relies on the development of novel order-explicit moment inequalities for higher-order Hoeffding components. We show that this rate is unimprovable, up to the polynomial factor on the logarithmic term. As corollaries, we obtain new Bernstein-type concentration and Gaussian approximation results for high-dimensional -statistics. We apply these results to establish the consistency of a set of resampling-based simultaneous confidence intervals built around a class of nonparametric regression estimators constructed with subsampled kernels. This class encompasses several forms of random forest regression, including Generalized Random Forests.