Berry–Esseen-type bounds to the normal, based on zeroand size-bias couplings, are derived using Stein’s method. The zero biasing bounds are illustrated in an application to combinatorial central limit theorems in which the random permutation has either the uniform distribution or one that is constant over permutations with the same cycle type, with no fixed points. The size biasing bounds are a...

A binary classification problem is considered, where the posteriori probability is estimated by the nonparametric kernel regression estimate with naive kernel. The excess error probability of the corresponding plug-in decision classification rule according to the error probability of the Bayes decision is studied such that the excess error probability is decomposed into approximation and estima...

*Correspondence: [email protected] 1School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, 530003, P.R. China Full list of author information is available at the end of the article Abstract In this paper, considering the nonparametric regression model Yni = g(ti) + εi (1≤ i≤ n), where εi =∑∞j=–∞ ajei–j and ei–j are identically distributed and ρ-mixing s...

Running head: Exponential Approximation by Exchangeable Pairs Version of 4/7/06 By Sourav Chatterjee and Jason Fulman Abstract: A general Berry-Esseen bound is obtained for the exponential distribution using Stein’s method of exchangeable pairs. As an application, an error term is derived for Hora’s result that the spectrum of the Bernoulli-Laplace Markov chain has an exponential limit. This is...

For some functions h (for example hz(w) = 1w≤z), the left-hand-side of (2) is a distance between a variable W and Z and we can bound that distance by taking advantage of the structures of h and W to bound the right-hand-side of (2). For example, by taking h such that the supremum of E[h(W )−Eh(Z)] yields the Wasserstein metric, and taking W = ∑ Xi/n for a sequence of independent mean-zero-varia...

It is well known that Gn(x) converges to the standard normal distribution function, (x), provided Eh(X1,X2) <∞ [see Hoeffding (1948)]. In fact, this moment condition can further be reduced to Eg(X1) <∞ and E|h(X1,X2)|4/3 <∞; see Remark 4.2.4 of Koroljuk and Borovskich [(1994), page 131]. In recent years, there has been considerable interest in obtaining rates of convergence in the asymptotic no...

We derive a Berry-Esseen bound, essentially of the order of the square of the standard deviation, for the number of maxima in random samples from (0, 1)d. The bound is, although not optimal, the first of its kind for the number of maxima in dimensions higher than two. The proof uses Poisson processes and Stein’s method. We also propose a new method for computing the variance and derive an asymp...

In 2001, Chen and Shao gave the nonuniform estimation of the rate of convergence in Berry-Esseen theorem for independent random variables via Stein-Chen-Shao method. The aim of this paper is to obtain a constant in Chen-Shao theorem, where the random variables are not necessarily identically distributed and the existence of their third moments are not assumed. The bound is given in terms of tru...

In this paper, we adapt the very effective Berry–Esseen theorems of Chen and Shao (2004), which apply to sums of locally dependent random variables, for use with randomly indexed sums. Our particular interest is in random variables resulting from integrating a random field with respect to a point process. We illustrate the use of our theorems in three examples; in a rather general model of the ...

In an attempt to investigate the adequacy of the normal approximation for the number of nuclei in certain growth/coverage models, we consider a Markov chain which has properties in common with related continuous-time Markov processes (as well as being of interest in its own right). We establish that the rate of convergence to normality for the number of \drops" during times 1; 2; : : : ; n is o...