论文标题
通过单数值收缩来适应一般二次损失
Adapting to general quadratic loss via singular value shrinkage
论文作者
论文摘要
高斯序列模型是非参数估计中的规范模型。在这项研究中，我们介绍了高斯序列模型的多元版本，并研究了多元Sobolev椭圆形的适应性估计，其中适应不仅对未知的平滑度，而且是任意二次损失。首先，我们通过Efron和Morris得出了奇异值收缩估计器的甲骨文不平等，这是James-Stein估计量的矩阵概括。接下来，我们针对每种二次损失开发了一个渐近的最小估计量，可以将其视为Pinsker定理的概括。然后，我们表明，在相应的二次损耗下，在多元Sobolev椭圆形上，块状efron- morris估计量完全是自适应的最小值。它同时获得了平均序列的任何线性组合的敏锐自适应估计。
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.
