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

Abstract

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.