Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 3, 588--610

COMBINING SAMPLE INFORMATION IN ESTIMATING ORDERED NORMAL MEANS

By

CONSTANCE VAN EEDEN

and

JAMES V. ZIDEK, University of British Columbia, Canada

SUMMARY. In this paper we answer a question concerned with the estimation of $\theta_1$ when $Y_i \sim^{ind}{\cal N}(\theta_i, \sigma_i^2), i=1, 2$, are observed and $\theta_1 \leq \theta_2$. In this case $\theta_2$ contains information about $\theta_1$ and we show how the relevance weights in the so-called weighted likelihood might be selected so that $Y_2$ may be used together with $Y_1$ for effective likelihood-based inference about $\theta_1$. Our answer to this question uses the Akaike entropy maximization criterion to find the weights empirically. Although the problem of estimating $\theta_1$ under these conditions has a long history, our estimator appears to be new. Unlike the MLE it is continuously differentiable. Unlike the Pitman estimator for this problem, but like the MLE, it has a simple form. The paper describes the derivation of our estimator, presents some of its properties and compares it with some obvious competitors. One of these competitors is the inadmissible maximum likelihood estimator for which we present a dominator. Finally, a number of open problems are presented.

*AMS (1991) subject classification}. *62F30, 62F10, 62C15, 62C20.

*Key words and phrases. *Likelihood, maximum likelihood, weighted likelihood, estimation, admissibility, minimaxity, normal means, restricted parameter spaces, relevance weighting.