Consistent estimation of the minimum normal mean under the tree-order restriction

Let (Xij∣j=1,…,ni(s),i=0,1,…,s) be independent observations from s+1 univariate normal populations, with Xij∼N(μi,σ2). The tree-order restriction (μ0⩽μi, i=1,…,s) arises naturally when comparing a treatment (μ0) to several controls (μ1,…,μs). When the sample sizes and population means and variances are equal and fixed, the maximum likelihood-based estimator (MLBE) of μ0 is negatively biased and diverges to -∞ a.s. as s→∞, leading some to assert that maximum likelihood may “fail disastrously” in order-restricted estimation. By viewing this problem as one of estimating a target parameter μ0 in the presence of an increasing number of nuisance parameters μ1,…,μs, however, this behavior is reminiscent of the classical Neyman-Scott example. This suggests an alternative formulation of the problem wherein the sample size n0(s) for the target parameter increases with s. Here the MLBE of μ0 is either consistent or admits a bias-reducing adjustment, depending on the rate of increase of n0(s). The consistency of an estimator due to Cohen and Sackrowitz [2002. Inference for the model of several treatments and a control. J. Statist. Plann. Inference 107, 89-101] is also discussed.