  Models for extreme values are generally derived from limit results, which are
meant to be good enough approximations when applied to finite samples.
Depending on the speed of convergence of the process underlying the data, these
approximations may fail to represent subasymptotic features present in the
data, and thus may introduce bias. The case of univariate maxima has been
widely explored in the literature, a prominent example being the slow
convergence to their Gumbel limit of Gaussian maxima, which are better
approximated by a negative Weibull distribution at finite levels. In the
context of subasymptotic multivariate extremes, research has only dealt with
specific cases related to componentwise maxima and multivariate regular
variation. This paper explores the conditional extremes model (Heffernan and
Tawn, 2004) in order to shed light on its finite-sample behaviour and to reduce
the bias of extrapolations beyond the range of the available data. We identify
second-order features for different types of conditional copulas, and obtain
results that echo those from the univariate context. These results suggest
possible extensions of the conditional tail model, which will enable it to be
fitted at less extreme thresholds.
