Sequential allocation for an estimation problem with ethical costs
type: Item
instance of: scholarly article
title: Sequential allocation for an estimation problem with ethical costs
zbMATH Open document ID: 0712.62074
published in: The Annals of Statistics
publication date: 1990-01-01T00:00:00Z
review text: This paper deals with the problem of designing an experiment to estimate the difference \(\theta\) of the means of two normal populations with unit variance. A quasi-Bayesian, decision-theoretic approach is adopted where \(\theta\) is estimated by its maximum likelihood estimator, but the design is evaluated in Bayesian, decision-theoretic terms. The risk function of a sequential design consists of the cost of drawing a sample from either population, which may depend on unknown \(\theta\), and the loss function of the form tK[\(\sqrt{t}(\theta -{\hat \theta})]\), where t is the sample size and K is a nonnegative, nonconstant, symmetric function of polynomial growth. An asymptotic expression for integrated risks is derived. An ad hoc three-stage procedure, which takes observations in three batches, is given. The regret of the three-stage procedure is shown to be bounded as \(t\to \infty\). A series of theorems indicate that the three-stage procedure is asymptotically second-order efficient for squared error loss. Some comments on conditions used in the analysis and suggestions for further investigations are also presented.
Mathematics Subject Classification ID: 62L12
Mathematics Subject Classification ID: 62L05
Mathematics Subject Classification ID: 62C10
zbMATH DE Number: 4172199
zbMATH Keywords: quasi-Bayesian approach
zbMATH Keywords: mean difference
zbMATH Keywords: sampling costs
zbMATH Keywords: invariance
zbMATH Keywords: posterior distributions
zbMATH Keywords: asymptotic normality
zbMATH Keywords: normal populations
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: risk function
zbMATH Keywords: loss function
zbMATH Keywords: asymptotic expression for integrated risks
zbMATH Keywords: three-stage procedure
zbMATH Keywords: regret
zbMATH Keywords: asymptotically second-order efficient
zbMATH Keywords: squared error loss
author: Michael Woodroofe
author: Janis P. Hardwick
reviewed by: Katsuji Uosaki
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176347754
OpenAlex ID: W2039720429
DOI: 10.1214/AOS/1176347754
description: scientific article; zbMATH DE number 4172199