Analysis of Functional Data without Dimension Reduction: Tests for Covariance Operators and Changepoint Problems

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Functional data arises in many applications and the main strategy for statistical inference is dimension reduction: The data is projected on a finite-dimensional space with techniques such as functional principal components. After this, it is possible to use statistical test for finite-dimensional data. In contrast, there are recent proposals to base the statistical tests on the full functional information, typically modeld as Hilbert-space-valued time series. These methods have been investigated in the context of sample means and simple changepoints.The aim of this project is to develop fully functional methods in more complicated data situations: We will investigate test for hypothesis not on the functional mean, but on the covariance operator. Furthermore, we plan to develop test for changepoints in data including extreme outliers, which might lead to false negatives and false positive results of standard methods. The last part will deal with segmentation of functional time series or detection of multiple changepoints. To get critical values, we will extend nonparametric methods like bootstrap to these challenging data situations.