The Grenander Estimator Under Model Misspecification

• Activity Number: 15
• Type: Topic Contributed
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Hanna Jankowski*+ and Jon Wellner
• Companies: York University and University of Washington
• Keywords: nonparametric ; misspecification ; Grenander ; maximum likelihood ; shape-constraints ; convergence rates

Abstract

Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate n^{-1/3} if the true density is curved and at rate n^{-1/2} if the density is flat. In the case that the true density is misspecified, the results of Patilea (2001) tell us that the global convergence rate is of order n^{-1/3}. We show that the local convergence rate is n^{-1/2} at a point where the density is misspecified. This is not in contradiction with the results of Patilea: the global convergence rate simply comes from locally curved well-specified regions. Furthermore, we study global convergence under misspecification by considering linear functionals. The rate of convergence is n^{-1/2} and we show that the limit is made up of two independent terms: a mean-zero Gaussian term and a second term (with non-zero mean) which is present only if the density has well-specified locally flat regions.