Multidimensional strong large deviation results
type: Item
instance of: scholarly article
title: Multidimensional strong large deviation results
zbMATH Open document ID: 1382.60054
author: Cyrille Joutard
published in: Metrika
publication date: 2017-12-01T00:00:00Z
review text: A strong large deviation theorem in the spirit of \textit{R. R. Bahadur} and \textit{R. Ranga Rao} [Ann. Math. Stat. 31, 1015--1027 (1960; Zbl 0101.12603)] is established for sequences \((\mathbf{Z}_n)\) of random vectors generalizing a similar theorem for the unidimensional case [\textit{C. Joutard}, Math. Methods Stat. 22, No. 2, 155--164 (2013; Zbl 1275.60031)]. Building on methods of proof by \textit{N. R. Chaganty} and \textit{J. Sethuraman} [Ann. Probab. 21, No. 3, 1671--1690 (1993; Zbl 0786.60026)], it is shown that under assumptions on the normalized cumulant generating functions (c.g.f.), the probability \(\mathbb{P}(\mathbf{Z}_n \geq \mathbf{a})\) can be approximated apart from a factor \((1+o(1))\) by an expression depending on the differentiable limit of the normalized c.g.f. and the \(n\)th element of the normalizing sequence. The theorem is applied to vectors of independent sampling variances, Gaussian multiple linear regression and the multivariate Nadaraya-Watson estimator. For the first two applications, the approximation is numerically compared to the true probabilities.
reviewed by: Philipp Doebler
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62G08
zbMATH DE Number: 6814688
zbMATH Keywords: large deviations
zbMATH Keywords: Bahadur-Rao theorem
zbMATH Keywords: sample variances
zbMATH Keywords: Gaussian multiple linear regression model
zbMATH Keywords: multivariate Nadaraya-Watson estimator
MaRDI profile type: Publication
full work available at URL: s00184-017-0621-4
OpenAlex ID: W2735896757
cites work: On Deviations of the Sample Mean
cites work: On Sharp Large Deviations for Sums of Random Vectors and Multidimensional Laplace Approximation
cites work: Sharp large deviations for Gaussian quadratic forms with applications
cites work: The Probability in the Extreme Tail of a Convolution
cites work: Q5514921
cites work: Integro-Local Limit Theorems Including Large Deviations for Sumsof Random Vectors.
cites work: Strong large deviation and local limit theorems
cites work: Multidimensional strong large deviation theorems
cites work: Q4391441
cites work: Q4692653
cites work: The bootstrap and Edgeworth expansion
cites work: Sharp asymptotics of large deviations in \(\mathbb{R}^ d\)
cites work: Sharp large deviations in nonparametric estimation
cites work: Strong large deviations for arbitrary sequences of random variables
cites work: Large and moderate deviations principles for kernel estimators of the multivariate regression
cites work: Dominating points and the asymptotics of large deviations for random walk on \(R^ r\).
cites work: On approximating probabilities for small and large deviations in \(R^{d1}\)
cites work: Multi-dimensional integral limit theorems for large deviations
cites work: A generalization of the Curtiss theorem for moment generating functions
DOI: 10.1007/S00184-017-0621-4
description: scientific article; zbMATH DE number 6814688