An Algorithm to Estimate Monotone Normal Means and its Application to Identify the Minimum Effective Dose

In the standard setting of one-way ANOVA with normal errors, a new algorithm, called the Step Down Maximum Mean Selection Algorithm (SDMMSA), is proposed to estimate the treatment means under an assumption that the treatment mean is nondecreasing in the factor level. We prove that i) the SDMMSA and the Pooled Adjacent Violator Algorithm (PAVA), a widely used algorithm in many problems, generate the same estimators for normal means, ii) the estimators are the mle's, and iii) the distribution of each of the estimators is stochastically nondecreasing in each of the treatment means. As an application of this stochastic ordering, a sequence of null hypotheses to identify the minimum effective dose (MED) is formulated under the assumption of monotone treatment(dose) means. A step-up testing procedure, which controls the experimentwise error rate in the strong sense, is constructed. When the MED=1, the proposed test is uniformly more powerful than Hsu and Berger's (1999).