Minimax rate adaptive estimation over continuous hyper-parameters

|We study minimax-rate adaptive estimation for density classes indexed by continuous hyper-parameters. The classes are assumed to be partially ordered in terms of inclusion relationship. Under a mild condition on the minimax risks, we show that a minimax-rate adaptive estimator can be constructed for the classes. 1 Problem of interest This paper concerns adaptive density estimation. Information-theoretic tools will be used to derive minimax-rate adaptive estimators for density classes indexed by continuous parameters. observations with density f(x); x 2 X with respect to a-nite measure. Here the space X is general and could be any dimensional. The goal is to estimate the unknown density f based on the data. The Kullback-Leibler (K-L) divergence between two densities f and g is deened as D(f k g) = R f log (f=g) dd: Let k k denote the L 2 norm with respect to the measure ; i.e., kgk = q R g 2 (x)(dx). Both the K-L loss D(f k ^ f) and the squared L 2 loss kf ? b fk 2 for an estimator b f will be considered for density estimation in this paper.