The Annals of Statistics

Convergence of linear functionals of the Grenander estimator under misspecification

Hanna Jankowski

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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.

Article information

Ann. Statist., Volume 42, Number 2 (2014), 625-653.

First available in Project Euclid: 20 May 2014

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Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G20: Asymptotic properties 62G07: Density estimation

Grenander estimator monotone density misspecification linear functional nonparametric maximum likelihood


Jankowski, Hanna. Convergence of linear functionals of the Grenander estimator under misspecification. Ann. Statist. 42 (2014), no. 2, 625--653. doi:10.1214/13-AOS1196.

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Supplemental materials

  • Supplementary material: Supplement to “Convergence of linear functionals of the Grenander estimator under misspecification”. We provide some proofs and technical details, as well as additional discussions of the assumptions in Theorems 3.1 and 4.1.