Bounds for the concentration of asymptotically normal statistics (Q1969255)

Let \(X_1,\dots, X_n\) be independent, not necessarily identically distributed r.v., taking values in a measurable space, and \({\mathbf T}= {\mathbf T} (X_1,\dots, X_n)\) be a real-valued statistic, with \(E{\mathbf T}^2 <\infty\). An upper bound is derived for the concentration function of \({\mathbf T}\) in terms of Hoeffding's decomposition for \({\mathbf T}\). The bound has a simple additive structure. Applications are given for Wilcoxon's rank-sum statistic, \(U\)-statistics, Student's statistics, the two-sample student statistic, and linear regression. In these applications, the exact knowledge of Hoeffding's decomposition is not needed, since instead of them the so-called differences are used; see \textit{V. Bentkus} et al., Ann. Stat. 25, No. 2, 851-896 (1997; Zbl 0920.62016). The main achievement of the paper under review is that the authors managed to drop the usual two assumptions of identical distributions of \(X_i\) and of the symmetry of \({\mathbf T}\). Note of the reviewer: The authors claim the optimality, up to absolute constants, of the obtained results. But the optimality is not proved in the paper. To show it, some additional considerations are necessary.