  Whether an extreme observation is an outlier or not, depends strongly on the
corresponding tail behaviour of the underlying distribution. We develop an
automatic, data-driven method to identify extreme tail behaviour that deviates
from the intermediate and central characteristics. This allows for detecting
extreme outliers or sets of extreme data that show less spread than the bulk of
the data. To this end we extend a testing method proposed in Bhattacharya et al
2019 for the specific case of heavy tailed models, to all max-domains of
attraction. Consequently we propose a tail-adjusted boxplot which yields a more
accurate representation of possible outliers. Several examples and simulation
results illustrate the finite sample behaviour of this approach.
