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This cumulative dissertation is organized as follows:
After the introduction, the second chapter, based on “Asymptotic independence of bivariate order statistics” (2017) by Falk and Wisheckel, is an investigation of the asymptotic dependence behavior of the components of bivariate order statistics. We find that the two components of the order statistics become asymptotically independent for certain combinations of (sequences of) indices that are selected, and it turns out that no further assumptions on the dependence of the two components in the underlying sample are necessary. To establish this, an explicit representation of the conditional distribution of bivariate order statistics is derived.
Chapter 3 is from “Conditional tail independence in archimedean copula models” (2019) by Falk, Padoan and Wisheckel and deals with the conditional distribution of an Archimedean copula, conditioned on one of its components. We show that its tails are independent under minor conditions on the generator function, even if the unconditional tails were dependent. The theoretical findings are underlined by a simulation study and can be generalized to Archimax copulas.
“Generalized pareto copulas: A key to multivariate extremes” (2019) by Falk, Padoan and Wisheckel lead to Chapter 4 where we introduce a nonparametric approach to estimate the probability that a random vector exceeds a fixed threshold if it follows a Generalized Pareto copula. To this end, some theory underlying the concept of Generalized Pareto distributions is presented first, the estimation procedure is tested using a simulation and finally applied to a dataset of air pollution parameters in Milan, Italy, from 2002 until 2017.
The fifth chapter collects some additional results on derivatives of D-norms, in particular a condition for the existence of directional derivatives, and multivariate spacings, specifically an explicit formula for the second-to-last bivariate spacing.
The Chaco leaf-cutting ant Atta vollenweideri (Forel) inhabits large and deep subterranean nests composed of a large number of fungus and refuse chambers. The ants dispose of the excavated soil by forming small pellets that are carried to the surface. For ants in general, the organisation of underground soil transport during nest building remains completely unknown. In the laboratory, we investigated how soil pellets are formed and transported, and whether their occurrence influences the spatial organisation of collective digging. Similar to leaf transport, we discovered size matching between soil pellet mass and carrier mass. Workers observed while digging excavated pellets at a rate of 26 per hour. Each excavator deposited its pellets in an individual cluster, independently of the preferred deposition sites of other excavators. Soil pellets were transported sequentially over 2 m, and the transport involved up to 12 workers belonging to three functionally distinct groups: excavators, several short-distance carriers that dropped the collected pellets after a few centimetres, and long-distance, last carriers that reached the final deposition site. When initiating a new excavation, the proportion of long-distance carriers increased from 18% to 45% within the first five hours, and remained unchanged over more than 20 hours. Accumulated, freshly-excavated pellets significantly influenced the workers' decision where to start digging in a choice experiment. Thus, pellets temporarily accumulated as a result of their sequential transport provide cues that spatially organise collective nest excavation.