On Variance Estimation for Tree Order Restricted Models

• Activity Number: 279
• Type: Contributed
• Date/Time: Tuesday, August 4, 2009 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Sanjay Chaudhuri*+ and Antar Bandyopadhyay
• Companies: National University of Singapore and Indian Statistical Institute
• Keywords: maximum likelihood estimation ; tree order restriction ; consistency

Abstract

We consider s+1 independent normal populations, with unknown means \mu_i, i=0,2,...,s and a common variance. We further assume that the means are constrained by a tree order restriction, namely \mu_0< =\mu_i, i=1,2,...,s. It is known that the constrained maximum likelihood estimator of \mu_0 is always biased and under very mild conditions may even diverge to -infinity as s increases to infinity. However conditions under which the same mle is bounded from below in probability or even consistent are also known. In this talk we consider estimation of the common variance under various such conditions. We show that as s tends to infinity, depending on the rate of growth of the sample sizes from the above populations with s, the mle of the common variance may sometimes be consistent and asymptotically normally distributed under appropriate centering and scaling.