Confidence Subset Containing the Unknown Peaks of an Umbrella Ordering

Abstract Umbrella ordering has received considerable attention in the statistical literature during the past 15 years. However, there is still no solution to the important problem of estimating the unknown peaks of an umbrella ordering. This article discusses confidence estimation of the unknown peaks. A random subset of all of the treatments that contains the unknown peaks of an umbrella ordering with any prespecified confidence level is considered. The proposed order-restricted confidence subset theory explicitly utilizes the order information and has several advantages for umbrella orderings over the usual subset selection theory. The utility and general applicability of the proposed method is shown by its applications in various experimental designs, assuming normal or other location-scale distributions with known or unknown variances. Examples based on real data are provided.