  In recent years several attempts have been made to extend tail modelling
towards the modal part of the data. Frigessi et al. (2002) introduced dynamic
mixtures of two components with a weight function {\pi} = {\pi}(x) smoothly
connecting the bulk and the tail of the distribution. Recently, Naveau et al.
(2016) reviewed this topic, and, continuing on the work by Papastathopoulos and
Tawn (2013), proposed a statistical model which is in compliance with extreme
value theory and allows for a smooth transition between the modal and tail
part. Incorporating second order rates of convergence for distributions of
peaks over thresholds (POT), Beirlant et al. (2002, 2009) constructed models
that can be viewed as special cases from both approaches discussed above. When
fitting such second order models it turns out that the bias of the resulting
extreme value estimators is significantly reduced compared to the classical
tail fits using only the first order tail component based on the Pareto or
generalized Pareto fits to peaks over threshold distributions. In this paper we
provide novel bias reduced tail fitting techniques, improving upon the
classical generalized Pareto (GP) approximation for POTs using the flexible
semiparametric GP modelling introduced in Tencaliec et al. (2018). We also
revisit and extend the secondorder refined POT approach started in Beirlant et
al. (2009) to all max-domains of attraction using flexible semiparametric
modelling of the second order component. In this way we relax the classical
second order regular variation assumptions.
