On the ....-Error of Monotonicity Constrained Estimators

We aim at estimating a function $\lambda \colon [0,1]\rightarrow {\Bbb R}$, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the ${\Bbb L}_{p}\text{-loss}$ of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of ., based on n observations. Our main task is to prove that the ${\Bbb L}_{p}\text{-loss}$ is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local ${\Bbb L}_{p}\text{-risk}$ at a fixed point and the global ${\Bbb L}_{p}\text{-risk}$ 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.