Statistica Sinica 24 (2014), 429-445

CONFIDENCE INTERVALS UNDER ORDER RESTRICTIONS

Yongseok Park$^1$, John D. Kalbfleisch$^2$ and Jeremy M. G. Taylor$^2$

$^1$University of Pittsburgh and $^2$University of Michigan

Abstract: In this paper, we consider the problem of constructing confidence intervals (CIs) for $G$ independent normal population means subject to linear ordering constraints. For this problem, CIs based on asymptotic distributions, likelihood ratio tests, and bootstraps do not have good properties, particularly when some of the population means are close to each other. We propose a new method based on defining intermediate random variables that are related to the original observations and using the CIs of the means of these intermediate random variables to restrict the original CIs from the separate groups. The coverage rates of the intervals are shown to exceed, but be close to, the nominal level for two groups, when the ratio of the variances is assumed known. Simulation studies show that the proposed CIs have coverage rates close to nominal levels with reduced average widths. Data on half-lives of an antibiotic are analyzed to illustrate the method.

Key words and phrases: Convex combination, elliptical unimodal distribution, linear ordering, normal distribution, restricted confidence interval.