Multiscale scanning with nuisance parameters
提出一种在干扰参数未知时对随机场进行多尺度扫描以发现异常区域的方法，通过在大尺度上估计干扰参数、仅用小尺度扫描，并证明调整后统计量的均匀不变性原理，可模拟渐近正确的临界值。
Abstract We develop a multiscale scanning method to find anomalies in a d-dimensional random field in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for multiscale scanning statistics will in general fail when such parameters are naively replaced by plug-in estimators. Instead, we suggest to estimate the nuisance parameters on the largest scale and to use (only) smaller scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic, which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate.