Biases of the maximum likelihood and Cohen-Sackrowitz estimators for the tree-order model

Consider s+1 univariate normal populations with common variance [sigma]2 and means [mu]i, i=0,1,...,s, constrained by the tree-order restrictions [mu]i[greater-or-equal, slanted][mu]0, i=1,2,...,s. For certain sequences [mu]0,[mu]1,... the maximum likelihood-based estimator (MLBE) of [mu]0 diverges to -[infinity] as s-->[infinity] and its bias is unbounded. By contrast, the bias of an alternative estimator of [mu]0 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 [mu]1,[mu]2,... are studied and compared to the biases of the corresponding Cohen-Sackrowitz estimators (CSE). Unlike the MLBE of [mu]0, the MLBEs of [mu]i for i[greater-or-equal, slanted]1, are asymptotically unbiased in most cases. By contrast, the CSEs of [mu]i, i=1,2,...,s more often have nonzero asymptotic bias.