Extremal and functional dependence between continuous random variables
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Date
2021
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Abstract
Describing and measuring the dependence between random variables is crucial to inform decisions regarding investments, policies, or even public safety. As the well-known coefficient of correlation cannot discriminate between independence and dependence in nonlinear settings, many other dependence concepts and corresponding measures of dependence have been proposed in the literature. This thesis investigates different dependence concepts for continuous random variables, while a special focus is placed on two particular dependence concepts, namely tail dependence and complete dependence.
In the first part of the thesis, we define the tail dependence ordering to consistently compare the degree of extremal dependence encoded in different sets of random variables and discuss corresponding measures of tail dependence, which include the well-known tail dependence coefficient as a special case. Furthermore, we provide conditions under which the tail dependence ordering is equivalent to a localized version of the usual stochastic dominance order. Afterwards, we investigate the tail behaviour of the (generalized) Markov product for copulas and introduce an analogous product structure for tail dependence functions. While this product is similar to the Markov product of copulas in several ways, for instance, regarding its algebraic properties or its connection to a class of linear operators, it exhibits a distinct reduction property allowing us to characterize the dynamic behaviour of n-fold iterates of the Markov product.
In the second part, we introduce a novel approach to rearrange a copula C into a unique stochastically increasing copula C↑. By combining measures of concordance such as Spearman's ρ or Kendall's τ with this rearrangement, we construct new measures of complete dependence. These so-called rearranged dependence measures are shown to possess various advantageous properties, e.g. they fulfil the data processing inequality, and are consistent with the underlying concordance measure. Lastly, we consider the theoretical properties of stochastically increasing copulas in more detail. This class of copulas exhibits improved convergence and order properties and includes various well-known copula families such as Gaussian or extreme-value copulas. Most importantly, stochastically increasing copulas fulfil a reduction property very similar to that of tail dependence functions, thus allowing us to characterize idempotents and n-fold iterates as ordinal sums of the independence copula.
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Copulas, Komplette Abhängigkeit, Randabhängigkeit