The Grenander estimator: A nonasymptotic approach. (English) Zbl 0703.62042

Birgé, Lucien

Ann. Stat. 17, No. 4, 1532-1549 (1989).

Summary: We shall investigate some nonasymptotic properties of the Grenander estimator of a decreasing density f. This estimator is defined as the slope of the smallest concave majorant of the empirical c.d.f. It will be proved that its risk, measured with \({\mathbb{L}}^ 1\)-loss, is bounded by some functional depending on f and the number n of observations. For classes of uniformly bounded densities with a common compact support, upper bounds for the functional are shown to agree with older results about the minimax risk over these classes. The asymptotic behavior of the functional as n goes to infinity is also in accordance with the known asymptotic performances of the Grenander estimator.

MSC:

62G05	Nonparametric estimation
60E15	Inequalities; stochastic orderings

Keywords:

local nonasymptotic risk; nonasymptotic properties; Grenander estimator of a decreasing density; uniformly bounded densities; minimax risk