Biases of the Maximum Likelihood and Cohen- Sackrowitz Estimators for the Tree-order Model

Consider s + 1 univariate normal populations with common variance σ2 and means μi, i = 0, 1, . . . , s, constrained by the tree-order restrictions μ0 ≤ μi, i = 1, 2, . . . , s. For certain sequences μ0, μ1, . . . the maximum likelihood-based estimator (MLBE) of μ0 diverges to −∞ as s → ∞ and its bias is unbounded. By contrast, the bias of an alternative estimator of μ0 proposed by Cohen and Sackrowitz (2002) remains bounded. In this note the biases of the MLBEs of the other components μ1, μ2, . . . are studied and compared to the biases of the corresponding Cohen-Sackrowitz estimators (CSE). Unlike the MLBE of μ0, the MLBEs of μi for i ≥ 1, are asymptotically unbiased in most cases. By contrast, the CSEs of μi, i = 1, 2, . . . , s more often have nonzero asymptotic bias.