Sequential detection of a change in a normal mean when the initial value is unknown
type: Item
instance of: scholarly article
title: Sequential detection of a change in a normal mean when the initial value is unknown
zbMATH Open document ID: 0732.62080
author: Moshe Pollak
author: David Siegmund
published in: The Annals of Statistics
publication date: 1991-01-01T00:00:00Z
review text: Let \(x_ 1,x_ 2,..\). be independent normal variables with variance 1 and E \(x_ n=\mu_ 0\) for \(n\leq \nu\) whereas E \(x_ n=\mu_ 0+\mu\) for \(n>\nu\), where \(\mu_ 0\) and \(\mu\) are unknown and of \(\nu\) it is only given that \(\nu \geq \nu_ 0\), a given nonnegative integer. The hypothesis \(\nu =\infty\) (meaning E \(x_ n=\mu_ 0\) for all n) is to be tested against the alternative \(\nu <\infty\). For any sequential procedure with stopping time T it is desirable to have \(E_{\infty}T\) large and \(E_{\nu}(T-\nu | T>\nu)\) small in some sense. At first it is assumed that \(\mu >0\). Three procedures are considered; all three are based on the maximal invariant \(y_ 2,y_ 3,...\), with \(y_ n=x_ n-x_ 1\), under the group of transformations \(x_ n\to x_ n+c\), \(n=1,2,...\), and the three corresponding stopping times are denoted N, \(\hat T,\) and \(\tau\), respectively. In the first procedure \(\delta >0\) and \(B>0\) are chosen and N is the least integer \(n>\nu_ 0\) for which \(R_ n(\delta)\geq B\), where \(R_ n(\delta)\) is the sum over i from \(\nu_ 0\) to n-1 of the likelihood ratio statistics (based on \(y_ 2,y_ 2,...)\) for testing \(\nu =\infty\) versus \(\nu =i\), \(\mu =\delta.\) In the second procedure \(R_ n(\delta)\) is replaced by \(\int R_ n(\delta)G(d\delta)\) with G a prior on \(\delta\). The third procedure is a CUSUM test based on the (invariant) sequence \(z_ 1,z_ 2,...\), where \(z_ n=(n/(n+1))^{1/2}(x_{n+1}-\bar x_ n).\) Two-sided analogues of these procedures are also considered in order to deal with arbitrary \(\mu \neq 0\). Asymptotic results are derived for \(E_{\infty}(T-\nu_ 0)\) as \(B\to \infty\) and for \(E_{\nu}(T-\nu | T>\nu)\) as \(B\to \infty\), \(\nu \to \infty\) under some restrictions, where \(T=N\) or \(\hat T.\) Monte Carlo experiments indicate that the asymptotic formulas provide reasonably good approximations. These experiments are also used to compare the three procedures. Their relative merits depend on the values of \…
Mathematics Subject Classification ID: 62L10
zbMATH DE Number: 4211317
zbMATH Keywords: stopping rules
zbMATH Keywords: mean
zbMATH Keywords: asymptotic approximations
zbMATH Keywords: average run length
zbMATH Keywords: training sample
zbMATH Keywords: change-point problem
zbMATH Keywords: sequential detection
zbMATH Keywords: independent normal variables
zbMATH Keywords: maximal invariant
zbMATH Keywords: group of transformations
zbMATH Keywords: likelihood ratio statistics
zbMATH Keywords: CUSUM test
zbMATH Keywords: Monte Carlo experiments
zbMATH Keywords: asymptotic formulas
reviewed by: Robert A. Wijsman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176347990
OpenAlex ID: W2059529543
DOI: 10.1214/AOS/1176347990
description: scientific article; zbMATH DE number 4211317