  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 [Sankhy\={a} Ser. A 31 (1969) 23-36] and at rate $n^{1/2}$ if the
density is flat [Ann. Probab. 11 (1983) 328-345; Canad. J. Statist. 27 (1999)
557-566]. In the case that the true density is misspecified, the results of
Patilea [Ann. Statist. 29 (2001) 94-123] 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 [Ann. Statist. 29
(2001) 94-123]: 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 nonzero mean) which is present
only if the density has well-specified locally flat regions.
