Comparing Groups With Umbrella Orderings
针对具有伞形排序（先升后降）的两组数据，开发了似然比检验来判断它们的峰值是否相同，并考虑了方差已知、未知且不同、未知且相同三种情况。
Abstract One commonly observed response pattern in a one-factor design with ordered treatment levels is the umbrella or unimodal ordering, in which the response variable increases with an increase in the treatment level up to a point, then decreases with further increase in the treatment level. The turning point is called the peak of an umbrella ordering. In many practical settings, there are two or more different populations with umbrella orderings, and it is of interest to compare these populations. In this article, likelihood ratio tests are developed for testing whether two groups with umbrella orderings have the same peaks. Three cases are considered: variances known, variances unknown and different for each group, and variances unknown and the same for both groups. Extensions to randomized block designs and to a more general class of testing problems are discussed.