Abstract:We give an order-explicit large deviation bound for the difference between a high-dimensional $U$-statistic and its Hájek projection. In particular, we show that any $U$-statistic of order $b$ on $n$ observations, with a $d$-dimensional kernel whose coordinates have $\psi_1$-Orlicz norm at most $\phi$, has a maximum deviation from its Hájek projection of order $O_p(\phi b n^{-1}\log^2(dn))$. 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 $U$-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.
From: David Ritzwoller [view email]
[v1]
Mon, 13 May 2024 15:46:11 UTC (225 KB)
[v2]
Fri, 23 Aug 2024 00:03:03 UTC (242 KB)
[v3]
Mon, 9 Sep 2024 18:33:18 UTC (243 KB)
[v4]
Wed, 13 May 2026 17:13:07 UTC (236 KB)
[v5]
Mon, 13 Jul 2026 19:52:34 UTC (240 KB)
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