On a converse to Scheffé's theorem
type: Item
instance of: scholarly article
title: On a converse to Scheffé's theorem
zbMATH Open document ID: 0605.62010
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: Let \(\{G_ n\}\) be a sequence of absolutely continuous distributions on \({\mathbb{R}}^ k\) and let \(\{g_ n\}\) be the corresponding sequence of densities. The main theorem says: The following two statements are equivalent: (1) \(\{g_ n\}\) is a.e.c. and bounded and \(G_ n\Rightarrow G.\) (2) \(g_ n\to g\) pointwise, uniformly in compacts of \({\mathbb{R}}^ k\) when g is the continuous density of the distribution G. a.e.c. stands for ''asymptotically equicontinuous'' what is defined as follows: for any \(x\in {\mathbb{R}}^ k\) and \(\epsilon >0\) there exist \(\delta =\delta (x,\epsilon)\) and \(n_ 0=n_ 0(x,\epsilon)\) such that whenever \(| y-x| \leq \delta\) then \(| g_ n(y)-g_ n(x)| <\epsilon\) for all \(n\geq n_ 0\) where \(| y-x|\) is the maximum absolute coordinate. This and the next similar result is an extension of an earlier result of \textit{D. D. Boos} [ibid. 13, 423-427 (1985; Zbl 0567.62012)]. An application to likelihood functions is also given.
reviewed by: Pál Révész
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 3978119
zbMATH Keywords: converse to Scheffé's theorem
zbMATH Keywords: convergence in distribution
zbMATH Keywords: uniform convergence of densities
zbMATH Keywords: uniform approximation of likelihood functions
zbMATH Keywords: asymptotically equicontinuous
author: Trevor Sweeting
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350065
OpenAlex ID: W2027915952
DOI: 10.1214/AOS/1176350065
description: scientific article; zbMATH DE number 3978119