Estimation of Order Restricted Normal Means when the Variances Are Unknown and Unequal

Abstract

<p dir="ltr">In the present paper, two normal distributions with parameters <span style="font-family: Calibri;"><span style="font-size: medium;">μ</span><sub><span style="font-size: small;">i </span></sub></span>and <span style="font-family: Calibri;"><span style="font-size: medium;">σ</span><span style="font-size: small;"><sub>i</sub><sup>2 </sup></span></span>where there is an order restriction on the means when the variances are unknown and unequal are considered. Under the squared error loss function, a necessary and sufficient condition for the plug-in estimators to improve upon the unrestricted maximum likelihood estimators uniformly is given. Also under the modified Pitman nearness criterion; a class of estimators is considered that reduce to the estimators of a common mean when the unbiased estimators violate the order restriction. It is shown that the most critical case for uniform improvement with regard to the unbiased estimators is the one when two means are equal. To illustrate the results, two numerical examples are presented.</p>