Convergence of linear functionals of the Grenander estimator under misspecification

TL;DR

This paper investigates the convergence behavior of the Grenander estimator under model misspecification, revealing local and global rates and the composition of the limiting distribution for linear functionals.

Contribution

It provides new insights into the local convergence rate of the Grenander estimator at misspecified points and characterizes the limit distribution of linear functionals under misspecification.

Findings

Local convergence rate at misspecified points is $n^{1/2}$.

Global convergence rate remains $n^{1/3}$ in Hellinger distance.

Limit distribution of linear functionals combines Gaussian and bias terms.

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 [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.