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 (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.