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The angular measure on the unit sphere characterizes the first-orderdependence structure of the components of a random vector in extreme regionsand is defined in terms of standardized margins. Its statistical recovery is animportant step in learning problems involving observations far away from thecenter. In the common situation that the components of the vector havedifferent distributions, the rank transformation offers a convenient and robustway of standardizing data in order to build an empirical version of the angularmeasure based on the most extreme observations. We provide a functionalasymptotic expansion for the empirical angular measure in the bivariate casebased on the theory of weak convergence in the space of bounded functions. Fromthe expansion, not only can the known asymptotic distribution of the empiricalangular measure be recovered, it also enables to find expansions and weaklimits for other statistics based on the associated empirical process or itsquantile version.

An asymptotic expansion of the empirical angular measure for bivariate extremal dependence