## The Annals of Statistics

### The Grenader Estimator: A Nonasymptotic Approach

Lucien Birge

#### Abstract

In this paper 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.

#### Article information

Source
Ann. Statist., Volume 17, Number 4 (1989), 1532-1549.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347380

Digital Object Identifier
doi:10.1214/aos/1176347380

Mathematical Reviews number (MathSciNet)
MR1026298

Zentralblatt MATH identifier
0703.62042

JSTOR