Adapting to General Quadratic Loss via Singular Value Shrinkage

The Gaussian sequence model is a canonical model in nonparametric estimation. In this study, 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 singular value shrinkage estimator by Efron and Morris, 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 the corresponding quadratic loss. It attains sharp adaptive estimation of any linear combination of the mean sequences simultaneously.