Below you find several topics I have worked on. Full texts of most of the papers can be found on arxiv.org.

 Change-point analysisWe develop tools for detecting changes in various characteristics of time series, such as the mean (Vogel & Wendler 2017), the variance (Gerstenberger, Vogel, Wendler 2020), or cross-sectional dependence (Dehling, Vogel, Wendler, Wied 2016; Vogel & Fried 2015). Classical tests for these purposes are based on moment estimates. These are not very well suited for heavy-tailed data, which, in many areas of applications, are rather the norm than the exception. The common theme of our work is an improved efficiency under heavy tails (actually moment-free) - while retaining the same performance under normality.

Graphical models

Graphical models provide a powerful tool to model the complex dependence structure of high-dimensional data. Conditional dependencies are represented by the edges of a graph, and graph-theoretic methods are employed in their analysis. The traditional working assumption is multivariate normality, which leads to the term “Gaussian graphical models” and allows a statistical inference based on the maximum-likelihood paradigm. We extend this to the semi-parametric class of elliptical distributions and show that graphical modelling can be based upon any covariance matrix estimator - as long as it satisfies two natural conditions: asymptotic normality and affine equivariance (Vogel & Fried 2011). We also propose Graphical M-estimators (Vogel & Tyler 2014) for robustly fitting graphical models. These work also for n < p. They are implemented in the R package robFitConGraph.

Music performance anxiety (MPA)

We investigate the origins of music performance anxiety (i.e., chronic stage fright): how it is related to childhood experiences and other types of anxieties. This is fundamental research aimed at finding effective therapies for MPA. And it is a nice application of a variaty of methods of multivariate statistics - graphical models being one of them. (Wiedemann et al. 2019; Wiedemann, Vogel, Voss, Hoyer 202?)

Robust high-dimensional correlation estimation: spherical correlation

The sample Pearson correlation matrix has a variety of very good statistical properties, among them: (1) it is guaranteed to be non-negative definite, (2) it is very fast to compute, and (3) it can be computed for n < p, i.e., whne the number of variables exceeds the number of observations.

However, it has one disadvantage: it does note cope well with heavy-tailed data. Alternative correlation matrix estimators that overcome that drawback usually fail at least one of the above three. Dürre, Fried, Vogel (2017) describe a correlation matrix estimator that is very robust wrt heay tails (and defined without any moment assumption) and possesses the three desirable properties above. This work is based on a series of earlier papers that lay the foundations: Dürre, Vogel, Tyler (2014); Dürre, Vogel, Fried (2015); Dürre, Vogel (2016); Dürre, Tyler, Vogel (2016). The methods are implemented in the R package sscor.

Scale measures

Gini's mean difference derives its name from its appearance in a 1912 paper by Corrado Gini. It is the enumerator of the Gini ratio and as such often used, but as a scale measure it has led much of a wallflower life in statistics. Maybe unfairly so: we show that it has very good statistical properties (Gerstenberger & Vogel 2016).

The distance standard deviation is another scale measure, which is related to the much acclaimed distance correlation. (Edelmann, Richards, Vogel 202?)

Multivariate location measures

The median is the robust counterpart to the mean - that much may be true in one dimension. In higher dimensions, several generalisations of the median exist (such as the componentwise median or the spatial median). Most of these lack one important property of the multivariate mean: affine equivariance, i.e., the mean of linearly transformed data is the thus transformed mean of the original data. This includes all scale changes and rotations. Hannu Oja (1983) was the first to suggest a multivariate median which is affine equivariant. The R package ojaNP contains several algorithms to conveniently compute the Oja median, exact algorithms as well as faster search heuristics. It also contains a lot of related content, such as Oja sign, ranks, and signed ranks, and estimates and tests based upon them. Its use is demonstrated in detail in Fischer et al. (2020).