The statistical information contained in additional observations
type: Item
instance of: scholarly article
title: The statistical information contained in additional observations
zbMATH Open document ID: 0633.62006
author: Enno Mammen
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: Within the setting of \textit{L. LeCam's} theory of comparison of experiments [Asymptotic methods in statistical decision theory. (1986; Zbl 0605.62002)] the pseudo-distance \(\Delta\) (\({\mathcal E}^ n,{\mathcal E}^{n+r_ n})\) between the powers of an experiment \({\mathcal E}\) defined by the measures \(P_{\theta}(\theta \in \Theta)\) are studied asymptotically as \(n\to \infty\). If \({\mathcal E}\) admits \(\sqrt{n}\)- constistent estimates, then \(\Delta\) (\({\mathcal E}^ n,{\mathcal E}^{n+r})\leq Cr/n\) for some constant \(C>0\). If \(\Theta \subset {\mathbb{R}}^ k\) and \({\mathcal E}\) can be locally approximated by a homoscedastic Gaussian experiment, then for \(r_ n=o(n)\) \textit{J. Helgeland}'s lower bound [Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 77-100 (1982; Zbl 0479.62006)] can be improved. Helgeland's upper bound turns out to be sharp. Tools for the proofs of the results are LeCam's randomization criterion for the deficiency \(\delta\) (\({\mathcal E},{\mathcal F})\) between experiments \({\mathcal E}\) and \({\mathcal F}\) and its representation in terms of the Bayes risks corresponding to \({\mathcal E}\) and \({\mathcal F}\), respectively.
Mathematics Subject Classification ID: 62B15
Mathematics Subject Classification ID: 62B10
zbMATH DE Number: 4030689
zbMATH Keywords: insufficiency
zbMATH Keywords: gain of information
zbMATH Keywords: additional observations
zbMATH Keywords: asymptotically Gaussian experiments
zbMATH Keywords: binomial family
zbMATH Keywords: pseudo-distance
zbMATH Keywords: powers of an experiment
zbMATH Keywords: constistent estimates
zbMATH Keywords: homoscedastic Gaussian experiment
zbMATH Keywords: LeCam's randomization criterion
zbMATH Keywords: deficiency
zbMATH Keywords: Bayes risks
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349945
OpenAlex ID: W2054159966
DOI: 10.1214/AOS/1176349945
description: scientific article; zbMATH DE number 4030689