On copula-based conditional dependency concepts and global dependency measures

Dependencies between two or more random variables can be influenced by some other random variables. Take for instance the dependency between male and female life expectancies at birth in various countries. This dependence turns out to be different in poor and wealthy countries. Such kind of dependencies that take the influence of a covariate on the dependency of the set of random variables into account are called conditional dependencies. Copulas are functions that characterize the dependence structure of a set of random variables. Association measures measure the strength of dependencies between a set of random variables. Examples include Spearman's rho and Kendall's tau. To study conditional dependencies, conditional copulas and conditional association measures are useful tools. When the interest is in studying conditional dependencies, and more precisely the strength of conditional dependencies, some kind of averaging over the conditioning random vector may be needed. Examples of average measures that can serve in this context are the average conditional Kendall's tau, which is the average of the conditional Kendall's tau with respect to the covariate, and the partial Kendall's tau. It is known that these measures differ in general. In the first part of the thesis we provide a quantitative study of the possible differences of the average conditional Kendall's tau and the partial Kendall's tau, and establish sufficient conditions under which they coincide. Both measures are studied in two fairly general settings. In each setting theoretical results are established as well as several illustrative examples are given. Another way of averaging over the conditioning random vector is by taking the median instead of the average (expectation) over the conditional copula or association measure with respect to the covariate vector. This leads to the notion of median conditional copulas and association measures. Since a median corresponds with the quantile of order 0.5, one could even study more generally quantile conditional copulas. In a second part of the thesis we investigate the existence of these concepts in three sub-general settings, and establish explicit expressions for calculating them. Examples are given to illustrate the concepts and estimators are proposed and evaluated through a simulation study. Also, the practical use of them is demonstrated in real data examples.

Jaar van publicatie:2021

Toegankelijkheid:Closed