On the double-struck L signp-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 double-struck L signp -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 L p-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local double-struck L signp-risk at a fixed point and the global double-struck L sign 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. © Institute of Mathematical Statistics, 2007.