On the LP-error of monotonicity constrained estimators

• 成果类型: Article
• 署名作者: Durot, Cecile
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000001497
• 发表日期: 2007
• 页码: 1080-1104
• 关键词: Asymptotic Normality; grenander-estimator; density

摘要

We aim at estimating a function lambda : [0, 1] -> R, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the L-p-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 Lp-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local L-p-risk at a fixed point and the global Lp-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.