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

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
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