  Extremal dependence describes the strength of correlation between the largest
observations of two variables. It is usually measured with symmetric dependence
coefficients that do not depend on the order of the variables. In many cases,
there is a natural asymmetry between extreme observations that can not be
captured by such coefficients. An example for such asymmetry are large
discharges at an upstream and a downstream stations on a river network: an
extreme discharge at the upstream station will directly influence the discharge
at the downstream station, but not vice versa. Simple measures for asymmetric
dependence in extreme events have not yet been investigated. We propose the
asymmetric tail Kendall's $\tau$ as a measure for extremal dependence that is
sensitive to asymmetric behaviour in the largest observations. It essentially
computes the classical Kendall's $\tau$ but conditioned on the extreme
observations of one of the two variables. We show theoretical properties of
this new coefficient and derive a formula to compute it for existing copula
models. We further study its effectiveness and connections to causality in
simulation experiments. We apply our methodology to a case study on river
networks in the United Kingdom to illustrate the importance of measuring
asymmetric extremal dependence in hydrology. Our results show that there is
important structural information in the asymmetry that would have been missed
by a symmetric measure. Our methodology is an easy but effective tool that can
be applied in exploratory analysis for understanding the connections among
variables and to detect possible asymmetric dependencies.
