Related papers: Bridging centrality and extremity: Refining empiri…
We consider the empirical versions of geometric quantile and halfspace depth, and study their extremal behaviour as a function of the sample size. The objective of this study is to establish connection between the rates of convergence and…
The halfspace depth is a well studied tool of nonparametric statistics in multivariate spaces, naturally inducing a multivariate generalisation of quantiles. The halfspace depth of a point with respect to a measure is defined as the infimum…
For measuring tail risk with scarce extreme events, extreme value analysis is often invoked as the statistical tool to extrapolate to the tail of a distribution. The presence of large datasets benefits tail risk analysis by providing more…
We study the empirical version of halfspace depths with the objective of establishing a connection between the rates of convergence and the tail behaviour of the corresponding underlying distributions. The intricate interplay between the…
Data depth is a statistical function that generalizes order and quantiles to the multivariate setting and beyond, with applications spanning over descriptive and visual statistics, anomaly detection, testing, etc. The celebrated halfspace…
A data depth measures the centrality of a point with respect to an empirical distribution. Postulates are formulated, which a depth for functional data should satisfy, and a general approach is proposed to construct multivariate data depths…
The use of expectiles in risk management has recently gathered remarkable momentum due to their excellent axiomatic and probabilistic properties. In particular, the class of elicitable law-invariant coherent risk measures only consists of…
Statistical depth is the act of gauging how representative a point is compared to a reference probability measure. The depth allows introducing rankings and orderings to data living in multivariate, or function spaces. Though widely applied…
With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications…
Depth is a concept that measures the `centrality' of a point in a given data cloud or in a given probability distribution. Every depth defines a family of so-called trimmed regions. For statistical applications it is desirable that with…
Statistical analysis is an important tool to distinguish systematic from chance findings. Current statistical analyses rely on distributional assumptions reflecting the structure of some underlying model, which if not met lead to problems…
Centrality descriptors are widely used to rank nodes according to specific concept(s) of importance. Despite the large number of centrality measures available nowadays, it is still poorly understood how to identify the node which can be…
In extreme value inference it is a fundamental problem how the target value is required to be extreme by the extreme value theory. In iid settings this study both theoretically and numerically compares tail estimators, which are based on…
The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step…
Modern statistical analyses often encounter datasets with massive sizes and heavy-tailed distributions. For datasets with massive sizes, traditional estimation methods can hardly be used to estimate the extreme value index directly. To…
Extreme value theory provides rigorous theory and statistical tools for extrapolation in machine learning, particularly in settings where traditional methods struggle due to data scarcity in the tails. A broad range of tasks benefit from…
A geometric representation for multivariate extremes, based on the shapes of scaled sample clouds in light-tailed margins and their so-called limit sets, has recently been shown to connect several existing extremal dependence concepts.…
Ranking or assessing centrality in multivariate and non-Euclidean data is difficult because there is no canonical order and many depth notions become computationally fragile in high-dimensional or structured settings. We introduce a…
Because it determines a center-outward ordering of observations in $\mathbb{R}^d$ with $d\geq 2$, the concept of statistical depth permits to define quantiles and ranks for multivariate data and use them for various statistical tasks (e.g.…
Although the fundamental probabilistic theory of extremes has been well developed, there are many practical considerations that must be addressed in application. The contribution of this thesis is four-fold. The first concerns the choice of…