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April 2014 Convergence of linear functionals of the Grenander estimator under misspecification
Hanna Jankowski
Ann. Statist. 42(2): 625-653 (April 2014). DOI: 10.1214/13-AOS1196


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.


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Hanna Jankowski. "Convergence of linear functionals of the Grenander estimator under misspecification." Ann. Statist. 42 (2) 625 - 653, April 2014.


Published: April 2014
First available in Project Euclid: 20 May 2014

zbMATH: 1302.62045
MathSciNet: MR3210981
Digital Object Identifier: 10.1214/13-AOS1196

Primary: 62E20 , 62G07 , 62G20

Keywords: Grenander estimator , linear functional , misspecification , monotone density , Nonparametric maximum likelihood

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 2 • April 2014
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