Estimation of the Larger of Two Normal Means

Publisher Website
Google Scholar
Abstract
Let Xi1, Xi2, ⋯, Xin, i = 1, 2, be a pair of random samples from populations which are normally distributed with means θi, and common known variance τ2. The problem is to estimate the function φ(θ1, θ2) = maximum (θ1, θ2). In this paper we consider five different estimators (or sets of estimators) for φ(θ1, θ2) and evaluate their biases and mean square errors. The estimators are (i) φ(X1, X2), where Xi is the sample mean of the ith sample; (ii) the analogue of the Pitman estimator, i.e. the a posteriori expected value of φ(θ1, θ2) when the generalized prior distribution is the uniform distribution on two dimensional space; (iii) a class of estimators which are generalized Bayes with respect to generalized priors which are products of uniform and normal priors; (iv) hybrid estimators, i.e. those which estimate by (X1 + X2)/2 when |X1 - X2| is small, and estimate by φ(X1, X2) when |X1 - X2| is large; (v) maximum likelihood estimator. The bias and mean square errors for these estimators are tabled, graphed, and compared. Also the invariance properties of these estimators are discussed with a rationale for restricting to invariant estimators.

This publication has 0 references indexed in Scilit: