A Nonparametric Test for Umbrella Alternate Scale Problem Using Ranked-set Data

One of the commonly observed one-factor response patterns in practice is the umbrella ordering in which the response variable increases with an increase in the treatment up to a point, then decreases with further increase in the treatment level. In this article, a nonparametric test for several sample scale problem against umbrella alternative with at least one strict inequality, when peak of the umbrella is known, is proposed using ranked-set data. The proposed statistic has the advantage of not requiring the several distribution functions to have a common median, but rather any common quantile of order , (not necessarily ) which is assumed to be known. The proposed test statistic is based on weighted linear combination of statistic proposed by Ozturk and Deshpande[11]. The distribution of the test statistic and optimal weights has been calculated. It is shown that the new test is uniformly more efficient in terms of Pitman asymptotic relative efficiency than its simple random sampling analog and its Mack-Wolfe (Mack and Wolfe (1981)) version as well.