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 [Sankhyā 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.
Citation
Hanna Jankowski. "Convergence of linear functionals of the Grenander estimator under misspecification." Ann. Statist. 42 (2) 625 - 653, April 2014. https://doi.org/10.1214/13-AOS1196
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