  The most popular approach in extreme value statistics is the modelling of
threshold exceedances using the asymptotically motivated generalised Pareto
distribution. This approach involves the selection of a high threshold above
which the model fits the data well. Sometimes, few observations of a
measurement process might be recorded in applications and so selecting a high
quantile of the sample as the threshold leads to almost no exceedances. In this
paper we propose extensions of the generalised Pareto distribution that
incorporate an additional shape parameter while keeping the tail behaviour
unaffected. The inclusion of this parameter offers additional structure for the
main body of the distribution, improves the stability of the modified scale,
tail index and return level estimates to threshold choice and allows a lower
threshold to be selected. We illustrate the benefits of the proposed models
with a simulation study and two case studies.
