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 μi⩾μ0, 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 (J. Statist. Plan. Infer. 107 (2002) 89-101) 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.