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This jump is clearly caused by the incompleteness of the MCXC catalogue and, thus, the inclusion of the missing clusters would most certainly move the observed sample to higher masses in the direction of the results we obtained from the SPT sample. In this sense, we do not see any indication of a substantial difference between the small- and wide-field survey. The analysis of the SPT sample illustrates the potential of utilizing the n most massive galaxy clusters to test underlying assumptions, e.

Order Statistic

In the upper panel of Fig. Of course, the observed masses might have to be corrected for the Eddington bias in mass and a possible reshuffling as previously demonstrated. Dependence of different percentiles on the order for the order statistics in mass upper panel and redshift lower panel. All percentiles are denoted by the same line styles as used in the previous figures and are given in the key. As before, we present in Fig.

Again, an increase in the survey area yields a shift of the theoretical distributions to higher redshifts and, as shown in the right-hand panel of Fig. Functional box plot for the first nine orders in the observable redshift as indicated in the individual panels. Here, the red line denotes the median Q 50 , the blue-bordered, grey region the IQR and the black lines denote the 2 and the 98 percentile Q 2, Q The definition of boxes and whiskers remains unchanged with respect to Fig. Thus, we abstained from the MC simulation of the reshuffling in the case of redshift.

While for the order statistics in mass the results depended only on the choice of the survey area, the situation is different for the order statistics in redshift. Here, a constant survey limiting mass is assumed, which will be subject to uncertainties for a real survey and, furthermore, will also exhibit some redshift dependence. Thus, the theoretical distributions are intrinsically less accurate than the ones with respect to cluster mass. Indeed, the comparison with the data in Fig.

Here, first four orders seem to be fitted better by the Tinker mass function while the higher orders seem to favour the PS mass function. However, considering the previously mentioned simplifying assumptions in the modelling of the theoretical distributions, we do not infer any cosmological conclusions and leave a better, more realistic, modelling of m lim z of the SPT survey to a future work. For each order, the red lines denote the median Q 50 , the blue-bordered, grey boxes give the IQR and the black whiskers mark the range between the 2 and 98 percentile Q 2, Q The green circles with error bars in the case of photometric redshifts represent the observed cluster redshifts.

In the lower panel of Fig.

The method presented in this work allows us to construct similar exclusion criteria for any kind of survey design. Having studied the individual order statistics in mass and redshift in the previous section, we turn now to the study of the joint distributions of the order statistics as introduced in Section 2.

The simplest case of a joint order distribution is two dimensional.

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07 Analysis of Algorithms: Order Statistics

In this case, the pdf and cdf are given by equations 6 and 7 , respectively. Starting with the joint pdf, we present in Fig. All calculations assume the full sky and the Tinker mass function. Joint pdf f r s x , y see equation 6 for the observable mass left-hand panel and redshift right-hand panel for different combinations of rank as indicated in the upper-left of each small panel.

The colour bar is set to range from 0 to the maximum of the joint pdf for each rank combination. An inspection of the different pdfs in Fig.

This means that it is more likely to find the two largest values close to each other at absolute values that are smaller than the extreme value statistics, which would imply for the maximum alone. Then, when moving to combinations of the first with higher orders first row , it can be seen that the peaks of the pdfs move away from the diagonal and that they extend to larger values for the larger observable. This indicates that it is more likely to find the two systems with a larger separation in the observable when the difference between the considered orders is larger.


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Accordingly, for higher order combinations lower rows , the peaks of the joint pdf move to smaller values of the observables. It should also be noted that the peaks steepen for higher order combinations, confining the pdfs to smaller and smaller regions in the observable plane. As an example, the first and second largest observations upper leftmost panel can be realized in much larger area than the sixth and eighth largest one lower rightmost panel.

Apart from the joint pdfs, it is also instructive to study the joint cdfs as presented in Fig. Thus, also the joint cdf confirms that the two largest masses do not exhibit any tension with the concordance cosmology. The same conclusion applies in the case of the joint distribution in redshift.

Bibliographic Information

By means of equation 8 these results can be extended to the n -dimensional case, allowing the formulation of a likelihood function of the ordered sample of the n most massive or highest redshift clusters. In this work, we studied the application of order statistics to the mass and redshifts of galaxy clusters and compared the theoretically derived distributions with observed samples of galaxy clusters. Our work extends previous studies that hitherto considered only the extreme value distributions in mass or redshift.

We introduce all relations necessary to calculate pdfs and cdfs of the individual and joint order statistics in mass and redshift. In particular, we find a steepening of the cdfs for higher order distributions with respect to the extreme value distribution of both mass and redshift. This steepening corresponds to a higher constraining power from distributions of the n largest observations.

The presented method extends previous works to include exclusion criteria based on the n most massive or n highest redshift clusters for a given survey set-up. On the contrary, when choosing the order statistics in redshift, focus is laid on haloes that stem from the highest possible redshifts. However, the calculations will require a model of the survey characteristics in the form of a limiting survey mass as a function of redshift.

By putting the emphasis on either the most massive or on the highest redshift clusters above a given mass limit, the order statistics is, e.


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In addition to the individual order statistics, we study as example case also the joint two order statistics. We find that for the combination of the largest and the second largest observation, it is most likely to find them to be realized with very similar values with a relatively broadly peaked distribution. In order to allow a quick estimation of the distributions of the order statistics, we provide in Appendix B fitting formulae for F x as a function of the survey area for the cases of mass and redshift.

In the case of the order statistics in redshift, the quality of the fits depends on the chosen m lim. After introducing the theoretical framework, we compared the theoretical distributions with actually observed samples of galaxy clusters that we ranked by the magnitude of the observables mass and redshift. The results of the comparison can be summarized as follows. In the case of the order statistics in mass, we compared the theoretical expectations for the 10 largest masses for the PS, the Tinker and the ST mass functions.

Distribution of order statistics from finite population

Assuming WMAP7 parameters, we find that the nominal and the Eddington bias corrected values for the observed masses favour the Tinker and the ST mass functions. When considering the possible bias due to a reshuffling of the ranks caused by the large error bars taking only the statistical errors into consideration , we find that the SPT sample is consistent with the ST and Tinker mass functions.

Though the ST mass function seems to provide a better match, the results are still inconclusive in view of the presence of systematic errors that might result in a common shift in mass to all orders. The constraints are expected to tighten considerably once the error bars of all objects are scaled down by combining several cluster observables in multiwavelength studies. In contrast to the ranking in mass, the order statistics of the SPT clusters in redshift is less well fit by the theoretical distributions based on the Tinker mass function. One explanation could be found in a non-standard cosmological evolution to which the order statistics in redshift is more sensitive.

However, it is more likely that a more precise modelling including the redshift dependence of the true limiting survey mass of SPT will account for the observed deviations. Instead of utilizing order statistics to perform exclusion experiments, it can also be used for consistency checks of the completeness of the observed sample and of the modelling of the survey selection function as indicated by the analysis of the MCXC mass and the SPT redshift samples.

We introduced a powerful theoretical framework which allows us to calculate the expected individual and joint distribution functions of the n largest masses or the n highest redshifts of galaxy clusters in a given survey area. This approach is more powerful than the extreme value statistics that focuses on the statistics of the single largest observation alone. As a proof of concept, we compared the theoretical distributions with observed samples of galaxy clusters.

However, data of sufficient quantity, uniformity and completeness is still sparse such that constraints are not particularly tight. This situation will most certainly improve in the near and intermediate future. Since the emphasis of this work lies on the introduction of the theoretical framework of order statistics and its application to galaxy clusters, we contended ourselves with a study of cluster masses and redshifts. Unfortunately, the mass of a galaxy cluster is not a direct observable and subject to large scatter and observational biases.

In a follow-up work, we intend to extend the formalism to direct observables, like for instance X-ray luminosities, and to include the scatter in the scaling relations into the theoretical distributions. In this appendix, we outline the derivation of the most important relations of the order statistics and some subtleties considering their implementation.

Description

Schematic for the derivation of f i x and f r s x , y. The differences are shown for three different ranks, the largest black lines , the fifth largest blue lines and the tenth largest green lines one. We present the results in Fig. B1 again as relative differences.