We investigate the problem to find anomalies in a $d$-dimensional random field via multiscale scanning 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 the multiscale scanning statistic will in general fail when naively such parameters are replaced by plug-in estimators. Opposed to this, we suggest to estimate the nuisance parameters on the largest scale and to use the remaining 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：我们研究了在存在干扰参数的情况下，通过多尺度扫描检测 $d$ 维随机场中的异常现象的问题。这涵盖了常见的场景，即基线水平或方差等附加参数未知，并需要从数据中估计。我们认为，当这些参数被朴素地替换为插件估计量时，当前确定多尺度扫描统计量渐近正确临界值的最先进方法通常会失效。与此相反，我们建议在最大尺度上估计干扰参数，并使用剩余尺度进行多尺度扫描。我们证明了由此产生的调整后多尺度统计量（AMS）满足一个统一的不变性原理，该原理具有广泛的适用性，并提供了一种计算上可行的方法来模拟渐近正确的临界值。我们通过模拟研究以及超分辨率STED显微镜的真实数据示例，展示了我们理论结果的实际意义。这使我们能够在受控族系误差率的前提下，通过预扫描识别样本内部感兴趣的区域。