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 and scale 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 any quadratic loss. It attains sharp adaptive estimation of any linear combination of the mean sequences simultaneously.

翻译：高斯序列模型是非参数估计的典型模型。 在此研究中, 我们引入了高斯序列模型的多变量版本, 并调查多变量 Sobolev ELLOTIP 的适应性估计, 适应性不仅指未知的平滑和规模, 也指任意的二次损失。 首先, 我们为 Efron 和 Morris 的单值缩水估计器得出一个星角不平等性。 这是 James- Stein 估测器的矩阵概括。 其次, 我们开发了一个多变量 Sobolev ELLOTID 的微缩鼠标估测器, 用于每个二次损失, 这可被视为 Pinsker 理论的概括性。 然后, 我们显示, 阻断的 Efron- Morris 估测器 是在任何二次损失下的多变量 Sobolev ellopiid 的微缩缩缩影。 它可以同时对平均序列的任何线性组合进行精确的调整性估计 。