Improved tests for comparing treatments against a control and other one-sided problems

Three multivariate “one-sided” testing problems are studied. All problems concern A: independent normal populations with unknown means and common unknown variance. For the first problem the null hypothesis is that all means are zero while the alternative is that each mean is nonnegative and at least one is positive. For the second problem the null hypothesis is that all means are equal and the alternative is that all of the first (k-1) means are greater than or equal to the mean of the last population (with strict inequality for at least one). The alternative for the second problem is called the “tree-top” alternative. For the third problem the null hypothesis is that all means are equal while the alternative is that the means are ordered, say nonincreasing. This latter alternative is called the “simple order” alternative. All three of these problems are discussed extensively in the book by Robertson, Wright, and Dykstra. We will assume that random samples of size n are drawn from each population although our results will hold if unequal sample sizes are drawn. For the first problem we will prove that tests of every size (with one exception) based on the maximum modulus statistic (properly normalized by the usual estimate of the standard deviation) are inadmissible. For the second problem we prove that all contrast type tests (normalized by the usual estimate of the standard deviation) are inadmissible for all sizes but one. For the third problem we prove that the one sided range test as well as others recommended in the literature are inadmissible. In all three problems better tests are offered and the amount of improvement is studied.