Jean-Claude Waizmann, Stefano Ettori, Lauro Moscardini
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
generalised Pareto distribution (GPD), which is closely related to the general
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.
View original:
http://arxiv.org/abs/1201.3526
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