Normal approximation under local dependence

By Louis H. Y. Chen, Qi-Man Shao. 2004

243 citations recorded by the source corpus.

We establish both uniform and nonuniform error bounds of the Berry–Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs couple Stein’s method with the concentration inequality approach.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3cb770f5d1953f7a28edc704fc1e2edebbcdf148

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

On adding a list of numbers (and other one-dependent determinantal processes) (2009, methodology): “(1) An O(n−1/2) error bound in this central limit theorem can be proved using the dependency graph approach to normal approximation by Stein’s method [13], [19].”
Stein’s method on Wiener chaos (2007, background): “Since Stein’s initial contribution [48], an impressive panoply of techniques has been developed in this direction (see again [10] or [43] for a survey; here, we shall quote e.”
A short survey of Stein's method (2014, background): “Stein’s method for a mixture of two normal distributions, with an application to spin glasses, appeared in my 2010 paper [23], while another non-normal distribution arising at the critical temperature of the Curie-Weiss model of ferromagnets was tackled in my joint paper with Shao [30] in 2011 and i”
Stein's method for normal approximation (2005, methodology): “The approach has further been shown to be effective in obtaining not only uniform but also non-uniform Berry–Esseen bounds for the accuracy of normal approximation for independent random variables (Chen & Shao, 2001) and for locally dependent random fields as well (Chen & Shao, 2004a).”
Mod-ϕ Convergence (2016, result): “For more general and in some cases sharper bounds, we also refer the reader to the work of Chen and Shao [CS04].”
Stein couplings for normal approximation (2010, methodology): “A typical instance is the use of higher-order neighbourhoods under local dependence as in Chen and Shao (2004), or the recursive structure in the combinatorial CLT in Bolthausen (1984).”
Fundamentals of Stein's method (2011, background): “It is also possible to prove similar theorems using further information about the dependence structure of the variables; see [25].”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “The reader is referred to [2] and [36], and the references therein, for an introduction to these techniques.”
Normal Approximation in Geometric Probability (2004, methodology): “Lemma 4.1 (see Thm 2.7 of [ 7 ]) Let 2 < q ≤ 3. Let Wi, i ∈ V, be random variables indexed by the vertices of a dependency graph.”
Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, unknown): “[24] Chen, L.”
Pareto Smoothed Importance Sampling (2015, methodology): “Importance sampling LOO was proposed by Gelfand et al. (1992), but it has not been widely used as the estimate is unreliable if the weights have infinite variance.”
Extended follow-up and spatial analysis of the American Cancer Society study linking particulate air pollution and mortality. (2009, background): “Under mild regularity conditions, the component-wise asymptotic normality of parameter estimator can be shown (He and Shao 2000; Chen and Shao 2004).”
A NEW METHOD OF NORMAL APPROXIMATION (2006, background): “, exchangeable pairs [41], diffusion generators [5], dependency graphs [4, 13, 36], size bias transforms [23], zero bias transforms [22], specialized procedures like [21, 34, 35] and recent developments [11, 12], to cite a few), but, somehow, they all require something”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, methodology): “Applying the Berry–Esseen theorem for dependency graphs (Chen and Shao, 2004, Theorem 2.7) to GDepn yields the desired result (7.8).”
Berry Esseen Bounds for Combinatorial Central Limit Theorems and Pattern Occurrences, using Zero and Size Biasing (2005, background): “[7], [19], [8], [24], [25], [10]) have been successful in obtaining bounds on the distance δ = sup h∈H |EI h(W )−Nh| (6)”