We aim at estimating a function λ:[0,1]→ℝ, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the $\mathbb {L}_{p}$ -loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of λ, based on n observations. Our main task is to prove that the $\mathbb {L}_{p}$ -loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local $\mathbb {L}_{p}$ -risk at a fixed point and the global $\mathbb {L}_{p}$ -risk are of order n−p/3. Applying the results to the density and regression models, we recover and generalize known results about Grenander and Brunk estimators. Also, we obtain new results for the Huang–Wellner estimator of a monotone failure rate in the random censorship model, and for an estimator of the monotone intensity function of an inhomogeneous Poisson process.
Publié le : 2007-07-14
Classification:
Central limit theorem,
drifted Brownian motion,
inhomogeneous Poisson process,
least concave majorant,
monotone density,
monotone failure rate,
monotone regression,
62G05,
62G07,
62G08,
62N02
@article{1185303999,
author = {Durot, C\'ecile},
title = {On the $\mathbb{L}\_{p}$ -error of monotonicity constrained estimators},
journal = {Ann. Statist.},
volume = {35},
number = {1},
year = {2007},
pages = { 1080-1104},
language = {en},
url = {http://dml.mathdoc.fr/item/1185303999}
}
Durot, Cécile. On the $\mathbb{L}_{p}$ -error of monotonicity constrained estimators. Ann. Statist., Tome 35 (2007) no. 1, pp. 1080-1104. http://gdmltest.u-ga.fr/item/1185303999/