Model selection for density estimation with -loss
成果类型：
Article
署名作者：
Birge, Lucien
署名单位：
Sorbonne Universite
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-013-0488-x
发表日期：
2014
页码：
533-574
关键词：
摘要：
We consider here estimation of an unknown probability density belonging to where is a probability measure. We have at hand i.i.d. observations with density and use the squared -norm as our loss function. The purpose of this paper is to provide an abstract but completely general method for estimating by model selection, allowing to handle arbitrary families of finite-dimensional (possibly non-linear) models and any . We shall, in particular, consider the cases of unbounded densities and bounded densities with unknown -norm and investigate how the -norm of may influence the risk. We shall also provide applications to adaptive estimation and aggregation of preliminary estimators. One major technical tool of our approach is a proof of the existence of suitable tests between -balls with centers belonging to . Although of a purely theoretical nature, our method leads to results that cannot presently be reached by more concrete ones.