Open Access
July 2007 On the $\mathbb{L}_{p}$-error of monotonicity constrained estimators
Cécile Durot
Ann. Statist. 35(3): 1080-1104 (July 2007). DOI: 10.1214/009053606000001497


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 np/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.


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Cécile Durot. "On the $\mathbb{L}_{p}$-error of monotonicity constrained estimators." Ann. Statist. 35 (3) 1080 - 1104, July 2007.


Published: July 2007
First available in Project Euclid: 24 July 2007

zbMATH: 1129.62024
MathSciNet: MR2341699
Digital Object Identifier: 10.1214/009053606000001497

Primary: 62G05 , 62G07 , 62G08 , 62N02

Keywords: central limit theorem , drifted Brownian motion , inhomogeneous Poisson process , least concave majorant , monotone density , monotone failure rate , monotone regression

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • July 2007
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