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

Abstract

We aim at estimating a function $\lambda :[0,1]\to \mathbb{R}$, 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 $\lambda$, 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.

Published at http://dx.doi.org/10.1214/009053606000001497 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)