Optimal Berry Esseen Research Articles

A Berry-Esseen bound with (almost) sharp dependence conditions

Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of n−1∕2 in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1, the left random walk on the general linear group and functions of linear processes.

Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ and denote by $$S_n:= g_n \cdots g_1$$ the associated random matrix product, where $$g_j$$ ’s are i.i.d.’s with law $$\mu $$ . We study statistical properties of random variables of the form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ where $$x \in \mathbb {P}^{d-1}$$ , $$\sigma $$ is the norm cocycle and u belongs to a class of admissible functions on $$\mathbb {P}^{d-1}$$ with values in $$\mathbb {R}\cup \{\pm \infty \}$$ . Assuming that $$\mu $$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $$\mathbb {R}$$ and Hölder continuous target functions on $$\mathbb {P}^{d-1}$$ . As particular cases, we obtain new limit theorems for $$\sigma (S_n,x)$$ and for the coefficients of $$S_n$$ .

Asymptotic behaviours for maximum likelihood estimator of drift parameter in α-Wiener bridge process

ABSTRACT In this paper, we study the asymptotic behaviour, including Cramér-type moderate deviations and optimal Berry-Esseen bounds for maximum likelihood estimator of drift parameter in α-Wiener bridge process. For statistical practice, a self-normalized version asymptotic result for the maximum likelihood estimator is also established. Numerical simulations show that our new established maximum likelihood estimator-based test statistic outperforms than the moment estimator-based test statistic and the likelihood ratio test statistic in literature.

Berry–Esseen bounds for generalized U-statistics

In this paper, we establish optimal Berry–Esseen bounds for the generalized U-statistics. The proof is based on a new Berry–Esseen theorem for exchangeable pair approach by Stein’s method under a general linearity condition setting. As applications, an optimal convergence rate of the normal approximation for subgraph counts in Erdos̈–Rényi graphs and graphon-random graph is obtained.

An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds

This paper is concerned with the rate of convergence of the distribution of the sequence {Fn/Gn}, where Fn and Gn are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of Fn/Gn. As a tool for our work, an Edgeworth expansion for the distribution of Fn/Gn, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs.

Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes

Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given 3 > p ≤ 4 3 > p \leq 4 moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metric W 1 W_1 , where a related, integrated characteristic turns out to be optimal. As an application, we establish novel weak Edgeworth expansion and CLTs in L p L^p and W 1 W_1 . As another application, we show that a large class of high dimensional linear statistics admits Edgeworth expansions without any smoothness constraints, that is, no non-lattice condition or related is necessary. In all results, the necessary weak-dependence assumptions are very mild. In particular, we show that many prominent dynamical systems and models from time series analysis are within our framework, giving rise to many new results in these areas.

An optimal Berry–Esseen type theorem for integrals of smooth functions

An optimal Berry–Esseen type theorem for integrals of smooth functions

Optimal Berry–Esseen bound for parameter estimation of SPDE with small noise

We investigate a rate of convergence on asymptotic normality of the maximum likelihood estimator (MLE) for parameter θ appearing in parabolic SPDEs of the form duϵ(t,x)=(A0+θA1)uϵ(t,x)dt+ϵdW(t,x),where A0 andA1 are partial differential operators, W is a cylindrical Brownian motion (CBM) and ϵ↓0. We find an optimal Berry–Esseen bound for central limit theorem (CLT) of the MLE. It is proved by developing techniques based on combining Malliavin calculus and Stein’s method.

An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ3-metric measuring the difference between expectations of sufficiently smooth functions, like |·|3, of a sum of independent random variables X 1,..., X n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X 1,..., X n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).

Optimal Berry–Esseen bound for statistical estimations and its application to SPDE

We consider asymptotically normal statistics of the form F n / G n , where F n and G n are functionals of Gaussian fields. For these statistics, we establish an optimal Berry–Esseen bound for the Central Limit Theorem (CLT) of the sequence F n / G n is φ ( n ) in the following sense: there exist constants 0 < c < C < ∞ such that c ≤ d Kol ( F n / G n , Z ) / φ ( n ) ≤ C , where d Kol ( F n , Z ) = sup z ∈ R ∣ Pr ( F n ≤ z ) − Pr ( Z ≤ z ) ∣ . As an example, we find an optimal Berry–Esseen bound for the CLT of the maximum likelihood estimators for parameters occurring in parabolic stochastic partial differential equations.

Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series

Given a stationary sequence , we are interested in the rate of convergence in the central limit theorem of the empirical quantiles and the empirical distribution function. Under a general notion of weak dependence, we show a Berry–Esseen result with optimal rate n−1/2. The setup includes many prominent time series models, such as functions of ARMA or (augmented) GARCH processes. In this context, optimal Berry–Esseen rates for empirical quantiles appear to be novel.

A Sufficient Condition on Optimal Berry-Esseen Bounds of Functionals of Gaussian Fields

We find a sufficient condition for optimal Berry-Esseen bounds in the normal approximation of functionals of Gaussian fields studied by Nourdin and Peccati (2009b).

Subtree Sizes in Recursive Trees and Binary Search Trees: Berry–Esseen Bounds and Poisson Approximations

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry–Esseen bounds and local limit theorems for the normal range and prove a Poisson approximation result as the subtree size tends to infinity.

Shortening the distance between Edgeworth and Berry–Esseen in the classical case

We obtain near optimal Berry–Esseen bounds for standardized sums of independent identically distributed random variables. This is achieved by distinguishing the lattice and the non-lattice cases, as one-term Edgeworth expansions do. The main tool is an easy inequality involving the usual second modulus of continuity, in substitution of Esseen's smoothing inequality. An illustrative example concerning the exponential distribution is also considered.

Normal Approximation Rate and Bias Reduction for Data-Driven Kernel Smoothing Estimator in a Semiparametric Regression Model

Accuracy of the normal approximation for Speckman's kernel smoothing estimator of the parametric component β in the semiparametric regression model y=xτβ+g(t)+e is studied when the bandwidth used in the estimator is selected by a general data-based method which includes such commonly used bandwidth selectors as (delete-one-out) CV, GCV, and Mallows' CL criterion. We find that, contrary to what we might expect, this data-driven estimator cannot attain the optimal Berry–Esseen rate n−1/2. Consequently, the confidence region of β based on this normal approximation is not first-order accurate. The reason for this is that the bias of Speckman's estimator is still of nonparametric order at the data-driven bandwidth choice. We then propose a resmoothing method to reduce the bias and show that the proposed estimator can achieve the optimal Berry–Esseen rate. A simulation study shows a slightly better small-sample performance of the proposed estimator.

Berry–Esseen bounds for von Mises and U-statistics

Let X1,... ,Xn be i.i.d.\ random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form ∑j> kH (Xj, Xk), where H is a measurable, symmetric function such that E | H (X1, X2)| > ∞, assuming that the statistic is non-degenerate. The same is done for von Mises statistics, that is, statistics of the form ∑j,kH (Xj, Xk). As a corollary, the central limit theorem is derived under optimal moment conditions.

A Berry–Esseen Bound for U-Statistics in the Non-I.I.D. Case

Let X 1,..., X n be independent, not necessarily identically distributed random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form T=∑1≤i<j≤n g ij(X i, X j), where the g ij are measurable functions such that $${\mathbb{E}}$$ |g ij(X i, X j)|<∞. An application is given concerning Wilcoxon's rank-sum test.

Normal Approximation Rate of the Kernel Smoothing Estimator in a Partial Linear Model

By establishing the asymptotic normality for the kernel smoothing estimatorβnof the parametric componentsβin the partial linear modelY=X′β+g(T)+ε, P. Speckman (1988,J. Roy. Statist. Soc. Ser. B50, 413–456) proved that the usual parametric raten−1/2is attainable under the usual “optimal” bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric componentg. In this paper we investigate the accuracy of the normal approximation forβnand find that, contrary to what we might expect, the optimal Berry–Esseen raten−1/2is not attainable unlessgis undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the “optimal” bandwidth choice.

Berry-Esséen rate in asymptotic normality for perturbed sample quantiles

In estimating quantiles with a sample of sizeN obtained from a distributionF, the perturbed sample quantiles based on a kernel functionk have been investigated by many authors. It is well known that their behaviour depends on the choices of “window-width”, saywN. Under suitable and reasonably mild assumptions onF andk, Ralescu and Sun (1993) have recently proven that limN→∞N1/4wN=0 is the necessary and sufficient condition for the asymptotic normality of the perturbed sample quantiles. In this paper, their rate of convergence is investigated. It turns out that the optimal Berry-Esseen rate ofO(N−1/2) can be achieved by choosing the window-width suitably, saywN=O(N−1/2). The obtained results, in addition to being explicit enough to verify the sufficient condition for the asymptotic normality, improve Ralescu's (1992) result of which the rate is of order (logN)N−1/2.