论文标题
统计反问题中的最佳正则假设检验
Optimal regularized hypothesis testing in statistical inverse problems
论文作者
论文摘要
假设的测试是数学统计学中精心研究的主题。最近，在反问题的背景下也解决了这个问题，在这种情况下，这种问题的数量不是直接访问的，而只有在（潜在的）操作员反转之后。在这项研究中，我们提出了一种在反问题中进行假设检验的正规化方法，即允许基础估计量（或测试统计量）偏置。在温和的源条件类型假设下，我们得出了一个具有规定级别$α$的测试家族，然后分析了如何从该家族中使用最大功率选择测试。作为一个主要结果，我们证明正规测试始终与（经典）未注册测试一样好。此外，使用凸优化的工具，我们通过最大化功率功能提供了自适应测试，然后通过几个数量级，在数值模拟中优于先前的未注册测试。
Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of Inverse Problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in Inverse Problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions we derive a family of tests with prescribed level $α$ and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.
