Estimating the intensity of a random measure by histogram type estimators

• 成果类型: Article
• 署名作者: Baraud, Yannick; Birge, Lucien
• 署名单位: Sorbonne Universite; Universite Paris Cite; Universite Cote d'Azur
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0126-6
• 发表日期: 2009
• 页码: 239-284
• 关键词: gaussian white-noise; model selection; Adaptive estimation; DENSITY-ESTIMATION; counting process; nonparametric-estimation; asymptotic equivalence; geometrizing rates; wavelet shrinkage; Poisson processes

摘要

The purpose of this paper is to estimate the intensity of some random measure N on a set X by a piecewise constant function on a finite partition of X. Given a (possibly large) family M of candidate partitions, we build a piecewise constant estimator (histogram) on each of them and then use the data to select one estimator in the family. Choosing the square of a Hellinger-type distance as our loss function, we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound. Moreover, the selection procedure leads to a final estimator satisfying some oracle-type inequality, with, as usual, a possible loss corresponding to the complexity of the family M. When this complexity is not too high, the selected estimator has a risk bounded, up to a universal constant, by the smallest risk bound obtained for the estimators in the family. For suitable choices of the family of partitions, we deduce uniform risk bounds over various classes of intensities. Our approach applies to the estimation of the intensity of an inhomogenous Poisson process, among other counting processes, or the estimation of the mean of a random vector with nonnegative components.