A0717
Title: Adapting to arbitrary quadratic loss via singular value shrinkage
Authors: Takeru Matsuda - RIKEN Center for Brain Science (Japan) [presenting]
Abstract: The Gaussian sequence model is a canonical model in nonparametric estimation. We introduce a multivariate version of the Gaussian sequence model and investigate adaptive estimation over the multivariate Sobolev ellipsoids, where adaptation is not only to unknown smoothness but also to arbitrary quadratic loss. First, we derive an oracle inequality for the Efron-Morris singular value shrinkage estimator, which is a matrix generalization of the James-Stein estimator. Next, we develop an asymptotically minimax estimator on the multivariate Sobolev ellipsoid for each quadratic loss, which can be viewed as a generalization of Pinsker's theorem. Then, we show that the blockwise Efron-Morris estimator is exactly adaptive minimax over the multivariate Sobolev ellipsoids under any quadratic loss. It attains sharp adaptive estimation of any linear combination of the mean sequences.
