论文标题
在噪声存在下，函数的自适应估计
Adaptive estimation of a function from its Exponential Radon Transform in presence of noise
论文作者
论文摘要
在本文中，我们提出了一种局部自适应策略，用于估算其指数ra transform（ERT）数据的函数，而没有事先了解要估计的功能的平滑度。我们构建了一个非参数内核类型估计器，并表明，对于包括宽阔的Sobolev规则性量表的一类功能，我们提出的策略遵循最小的最佳最佳速率，最高为$ \ log {n} $ factor。我们还表明，当使用尖端风险时，在Sobolev量表上不存在最佳的自适应估计器，实际上，所提出的估计器所达到的速率是收敛的自适应速率。
In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a $\log{n}$ factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.
