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Gaussian approximations for high-dimensional non-degenerate \(U\)-statistics via exchangeable pairs. (English) Zbl 1478.60086

Summary: In this paper, we obtain a non-asymptotic bound for Gaussian approximations for centered high-dimensional non-degenerate \(U\)-statistics over the class of hyperrectangles via exchangeable pairs and Stein’s method. We improve the upper bound of the convergence rate from \(n^{-1/6}\) in [X. Chen, Ann. Stat. 46, No. 2, 642–678 (2018; Zbl 1396.62019)] to \(n^{- 1/4}\) up to a polynomial factor of \(\log d\) under the same conditions, where \(n\) is the sample size and \(d\) is the dimension of the \(U\)-statistic. Convergence to zero of the bound requires \(\log d = o (n^{1/7})\) in [loc. cit.], this requirement on \(d\) is weaken in this paper by allowing \(\log d = o (n^{1/5})\).

MSC:

60F05 Central limit and other weak theorems
62E17 Approximations to statistical distributions (nonasymptotic)
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 1396.62019
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References:

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