This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex, and closed model set containing the object. It complements the article "Statistical Estimation and Optimal Recovery" published in the Annals of Statistics in 1994. There, Donoho showed (among other things) that, for Gaussian noise, linear maps provide near-optimal estimation schemes relatively to a performance measure relevant in Statistical Estimation. Here, we advocate for a different performance measure arguably more relevant in Optimal Recovery. We show that, relatively to this new measure, linear maps still provide near-optimal estimation schemes even if the noise is merely log-concave. Our arguments, which make a connection to the deterministic noise situation and bypass properties specific to the Gaussian case, offer an alternative to parts of Donoho's proof.

翻译：本文探讨了通过线性观测（受随机噪声污染）获取的对象，其线性泛函的最优估计问题，其中最优性取决于包含该对象的对称、凸且闭的模型集所对应的最坏情况设定。本文是对1994年发表在《统计年鉴》上的《统计估计与最优恢复》一文的补充。在那篇文章中，Donoho 证明了（除其他结论外）对于高斯噪声，线性映射相对于统计估计中相关的性能度量提供了近最优的估计方案。在此，我们主张采用另一种性能度量，该度量在最优恢复中可能更具相关性。我们证明，相对于这一新度量，即使噪声仅具有对数凹性，线性映射仍能提供近最优的估计方案。我们的论证建立了与确定性噪声情形的联系，并绕开了高斯情况特有的性质，从而为 Donoho 证明的部分内容提供了一种替代方案。