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 n1/3 if the true density is curved [Sankhy. Ser. A 31 (1969) 23-36] and at rate n1/2 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 n1/3 in Hellinger distance. Here, we show that the local convergence rate is n1/2 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 n1/2 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.