# ESTIMATION OF THE MINIMUM MEAN OF NORMAL POPULATIONS UNDER THE TREE ORDER RESTRICTION

Authors: Sanjay Chaudhuri (Department of Statistics and Applied probability, National University of Singapore), Michael Perlman (Department of Statistics, University of Washington, Seattle)

### Abstract

We consider $k+1$ univariate normal populations with common variance $\sigma^2$ and means $\mu_i$, $i=0,1,2,\ldots,k$, constrained by the tree order restriction $\mu_0\le\mu_i$, $i=1,2,\ldots,k$. The maximum likelihood estimator of $\mu_0$ is known to diverge to $-\infty$ a.s. as $k\to\infty$ if all means $\mu_i$ are bounded and all sample sizes $n_i$ remain finite. However, this is not true if the means are unbounded or more importantly if $n_0$ increases with $k$, which is the case of most practical interest. In such cases it can be shown that the m.l.e. of $\mu_0$ is consistent, or atleast bounded from below. The consistency of a modified version of a estimator due to Cohen and Sackrowitz $(2002)$ also will be discussed.