Confidence Interval Estimation Subject to Order Restrictions

Abstract

This article deals with the construction of confidence intervals when the components of the location parameter $\mu$ of the random variable $\mathbf{X}$, which is elliptically symmetrically distributed, are subject to order restrictions. Several domination results are proved by studying the derivative of the coverage probability of the confidence intervals centered at the improved point estimators. Consequently, we strengthen the previously known results regarding the simple ordering and obtain several new results for other general forms of order restrictions, including the simple tree ordering, the umbrella ordering, the simple and the double loop ordering and some combination of these. These domination results are obtained under the assumption that $\Sigma$ is a diagonal matrix. When $\Sigma$ is nondiagonal, some new intervals are introduced which dominate the standard intervals centered at the unrestricted maximum likelihood estimator for various types of order restrictions. Similar results are obtained for scale parameters as well. Contrary to the location problems, in case of the scale parameters satisfying the simple ordering we find that the restricted maximum likelihood estimator of the largest parameter fails to universally dominate the unrestricted maximum likelihood estimator. A similar negative result is noted for simple tree order restriction.