Multiscale Scanning in Higher Dimensions: Limit theory, statistical consequences and an application in STED microscopy

by Claudia Juliane König
Doctoral thesis
Date of Examination:2018-06-26
Date of issue:2018-07-17
Advisor:Dr. Frank Werner
Referee:Dr. Frank Werner
Referee:Prof. Dr. Axel Munk
Persistent Address: http://dx.doi.org/10.53846/goediss-6968

 

 

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Abstract

English

Scan 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
 
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