We establish a Berry–Esseen bound for general multivariate nonlinear statistics by developing new multivariate-type randomized concentration inequality. The is the best possible many known statistics. As applications, bounds M-estimators and averaged stochastic gradient descent algorithms are obtained.

We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where noise matrix is allowed to have dependence within rows and heteroskedasticity between them. prove finite-sample $\ell _{2,\infty }$ bounds a Berry-Esseen theorem individual entries estimator...

Abstract Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds estimator. The results are believed to optimal or near-optimal.

We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities an improved Berry-Esseen inequality for Rademacher sums.

In this paper, we generalize the asymptotic result of Esseen (1958) concerning the Wasserstein distance of order one in the mean central limit theorem to the Wasserstein distances of order r for r in ]1, 2].

* Correspondence: lym1019@163. com Department of Mathematics, Shangrao Normal University, Shangrao 334001, China Full list of author information is available at the end of the article Abstract Consider the semiparametric regression model Yi = xi b + g (ti) + εi, i = 1, . . . , n, where the linear process errors εi = ∑∞ j=−∞ ajei−j with ∑∞ j=−∞ ∣∣aj∣∣ < ∞ , and {ei} are identically distributed a...