On the modelling of the excesses of galaxy clusters over high-mass
  thresholds
  In this work we present for the first time an application of the Pareto
approach to the modelling of the excesses of galaxy clusters over high-mass
thresholds. The distribution of those excesses can be described by the
generalized Pareto distribution (GPD), which is closely related to the
generalized extreme value (GEV) distribution. After introducing the formalism,
we study the impact of different thresholds and redshift ranges on the
distributions, as well as the influence of the survey area on the mean excess
above a given mass threshold. We also show that both the GPD and the GEV
approach lead to identical results for rare, thus high-mass and high-redshift,
clusters. As an example, we apply the Pareto approach to ACT-CL J0102-4915 and
SPT-CL J2106-5844 and derive the respective cumulative distribution functions
of the exceedance over different mass thresholds. We also study the possibility
to use the GPD as a cosmological probe. Since in the maximum likelihood
estimation of the distribution parameters all the information from clusters
above the mass threshold is used, the GPD might offer an interesting
alternative to GEV-based methods that use only the maxima in patches. When
comparing the accuracy with which the parameters can be estimated, it turns out
that the patch-based modelling of maxima is superior to the Pareto approach. In
an ideal case, the GEV approach is capable to estimate the location parameter
with a percent level precision for less than 100 patches. This result makes the
GEV based approach potentially also interesting for cluster surveys with a
smaller area.
