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

Abstract

Let ( X ij ∣ j = 1 , … , n i ( s ) , i = 0 , 1 , … , s ) be independent observations from s + 1 univariate normal populations, with X ij ∼ N ( μ i , σ 2 ) . The tree-order restriction ( μ 0 ⩽ μ i , i = 1 , … , s ) arises naturally when comparing a treatment ( μ 0 ) to several controls ( μ 1 , … , μ s ). When the sample sizes and population means and variances are equal and fixed, the maximum likelihood-based estimator (MLBE) of μ 0 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 μ 0 in the presence of an increasing number of nuisance parameters μ 1 , … , μ s , however, this behavior is reminiscent of the classical Neyman–Scott example. This suggests an alternative formulation of the problem wherein the sample size n 0 ( s ) for the target parameter increases with s. Here the MLBE of μ 0 is either consistent or admits a bias-reducing adjustment, depending on the rate of increase of n 0 ( 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.