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. Opposed to this, 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 (AMS), 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.

翻译：本文提出一种在存在干扰参数的情况下检测$d$维随机场中异常的多尺度扫描方法。这涵盖了基线水平或方差等附加参数未知且需从数据中估计的常见情形。我们论证指出，当这些参数被朴素地替换为插件估计量时，当前用于确定多尺度扫描统计量渐近正确临界值的先进方法通常将失效。与此相反，我们建议在最大尺度上估计干扰参数，并（仅）使用较小尺度进行多尺度扫描。我们为由此得到的调整后多尺度统计量（AMS）证明了具有广泛适用性的统一不变性原理，这为模拟渐近正确临界值提供了计算可行的方法。通过仿真研究以及来自超分辨率STED显微镜的真实数据示例，我们展示了理论结果的实际意义。这使得我们能够在控制族系误差率的预扫描中识别样本内的感兴趣区域。