The Annals of Statistics

On the $\mathbb{L}_{p}$-error of monotonicity constrained estimators

Cécile Durot

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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.

Article information

Ann. Statist., Volume 35, Number 3 (2007), 1080-1104.

First available in Project Euclid: 24 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G07: Density estimation 62G08: Nonparametric regression 62N02: Estimation

Central limit theorem drifted Brownian motion inhomogeneous Poisson process least concave majorant monotone density monotone failure rate monotone regression


Durot, Cécile. On the $\mathbb{L}_{p}$-error of monotonicity constrained estimators. Ann. Statist. 35 (2007), no. 3, 1080--1104. doi:10.1214/009053606000001497.

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