Multiscale Scanning in Higher Dimensions: Limit theory, statistical consequences and an application in STED microscopy
von Claudia Juliane König
Datum der mündl. Prüfung:2018-06-26
Betreuer:Dr. Frank Werner
Gutachter:Dr. Frank Werner
Gutachter:Prof. Dr. Axel Munk
EnglischScan statistics have a broad area of applications ranging from astrophysics over genetic screening to fluorescence microscopy. Here, we consider a calibrated scan statistic based on local likelihood ratio tests of homogeneity against heterogeneity. The problem is to find anomalies in a d-dimensional field of independent random variables, each distributed according to a one-dimensional natural exponential family. This thesis provides a unified methodology which controls the overall family wise error rate (FWER) to make a wrong detection at a given level. Fundamental to our method is a Gaussian approximation of the asymptotic distribution of the underlying multiscale scanning test statistic with explicit rate of convergence. From this, we obtain a weak limit theorem which can be seen as a generalized weak invariance principle to non-identically distributed data and is of independent interest. Furthermore, we give an asymptotic expansion of the procedure's power, which yields minimax optimality in case of Gaussian observations.
Keywords: multiscale testing; scan statistic; exponential families; invariance principle; weak limit; family wise error rate