
Statistics Papers
Document Type
Journal Article
Date of this Version
1993
Publication Source
Annals of Statistics
Volume
21
Issue
2
Start Page
625
Last Page
644
DOI
10.1214/aos/1176349141
Abstract
Consider the problem of estimating μ, based on the observation of Y0,Y1,…,Yn, where it is assumed only that Y0,Y1,…,YκiidN(μ,σ2) for some unknown κ. Unlike the traditional change-point problem, the focus here is not on estimating κ, which is now a nuisance parameter. When it is known that κ=k, the sample mean Y¯k=∑k0Yi/(k+1), provides, in addition to wonderful efficiency properties, safety in the sense that it is minimax under squared error loss. Unfortunately, this safety breaks down when κ is unknown; indeed if k>κ, the risk of Y¯k is unbounded. To address this problem, a generalized minimax criterion is considered whereby each estimator is evaluated by its maximum risk under Y0,Y1,…,YκiidN(μ,σ2) for each possible value of κ. An essentially complete class under this criterion is obtained. Generalizations to other situations such as variance estimation are illustrated.
Keywords
change-point problems, equivariance, Hunt-Stein theorem, minimax procedures, risk, pooling data
Recommended Citation
Foster, D. P., & George, E. I. (1993). Estimation up to a Change-Point. Annals of Statistics, 21 (2), 625-644. http://dx.doi.org/10.1214/aos/1176349141
Date Posted: 27 November 2017
This document has been peer reviewed.