Asymptotic local minimaxity in sequential point estimation
type: Item
instance of: scholarly article
title: Asymptotic local minimaxity in sequential point estimation
zbMATH Open document ID: 0581.62067
published in: The Annals of Statistics
publication date: 1985-01-01T00:00:00Z
review text: Let \(\{X_ n\), \(n\geq 1\}\) be a sequence of i.i.d. random variables with mean \(\theta\) and finite variance \(\sigma^ 2\), depending on unknown parameter \(\omega\in \Omega\). The author describes a stopping time t for which the risk \[ R_ A(t,\omega)=E_{\omega}\{A\gamma^ 2_ 0(\omega)(\bar X_ t-\theta)^ 2+t\} \] is as small as possible, where \(A>0\), \(\gamma_ 0\) is a positive function on \(\Omega\), and \(\bar X_ t=(X_ 1+...+X_ t)/t\). If \(t=n\) is a fixed sample size, then \(2\sqrt{A}(\gamma_ 0\sigma)\) is a lower bound for \(R_ A(n,\omega)\), \(n\geq 1\), and the regret of a stopping time t is defined to be \[ r_ A(t,\omega)=R_ A(t,\omega)-2\sqrt{A}(\gamma_ 0\sigma). \] The main results determine an asymptotic lower bound, as \(A\to \infty\), for the minimax regret \(M_ A(\Omega_ 0)=\inf_{t}\sup_{\omega \in \Omega_ 0}r_ A(t,\omega)\) for neighbourhoods \(\Omega_ 0\) of arbitrary parameter points \(\omega_ 0\in \Omega\). The bound is obtained for multiparameter exponential families and the nonparametric case.
Mathematics Subject Classification ID: 62L12
Mathematics Subject Classification ID: 62L15
Mathematics Subject Classification ID: 62F10
zbMATH DE Number: 3930175
zbMATH Keywords: weighted squared error loss
zbMATH Keywords: exponential families
zbMATH Keywords: nonparametric case
zbMATH Keywords: regret
zbMATH Keywords: Bayes risk
zbMATH Keywords: minimax theorem
zbMATH Keywords: martingale convergence
zbMATH Keywords: asymptotic lower bound
zbMATH Keywords: minimax regret
zbMATH Keywords: multiparameter exponential families
author: Michael Woodroofe
reviewed by: Zdzisław Rychlik
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349547
OpenAlex ID: W1987728095
DOI: 10.1214/AOS/1176349547
description: scientific article; zbMATH DE number 3930175