Convergence of Linear Functionals of the Grenander Estimator under Misspecification

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 (Prakasa Rao, 1969) and at rate n−1/2 if the density is flat (Groeneboom and Pyke, 1983; Carolan and Dykstra, 1999). 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 in Hellinger distance. Here, 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 (2001): 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.