Estimating the K-function of a point process with an application to cosmology

The Annals of Statistics (Impact Factor: 2.18). 07/2000; 28(6). DOI: 10.1214/aos/1015957468
Source: arXiv


Motivated by the study of an important data set for understanding the large-scale structure of the universe, this work considers the estimation of the reduced second moment function, or K-function, of a stationary point process observed over a large number of segments of possibly varying lengths. Theory and simulation are used to compare the behavior of isotropic and rigid motion correction estimators and some modifications of these estimators. These results generally support the use of modified versions of the rigid motion correction. When applied to a catalog of astronomical objects known as absorbers, the proposed methods confirm results from earlier analyses of the absorber catalog showing clear evidence of clustering up to 50 Mpc and marginal evidence for clustering of matter on spatial scales beyond 100 Mpc, which is beyond the distance at which clustering of matter is now generally accepted to exist.

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Available from: Jean Quashnock, Jun 19, 2013
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    • "They showed that the modii ed versions of the rigid motion estimators achieve the greatest reduction in standard errors compared to the unmodii ed estimators. Here we give an adaptation of the modi cations suggested in Stein et al. (2000) "
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    ABSTRACT: Galaxies have long been known to form large clusters, and cosmologists are interested in characterizing this clustering as a way of studying the large-scale structure of the universe. This work is motivated by a data catalog consisting of information on lines of sight from Earth to distant quasi-stellar objects (QSO's) and the carbon IV absorbers that lie on them. The absorbers are believed to be gas clouds near galaxies too far away to be easily observed. Thus, the absorber catalog provides a unique and interesting way to examine the large-scale structure of the universe. On large scales previous studies have mainly used pairs of absorbers on the same lines of sight to obtain estimates describing the clustering of absorbers. It is clear that absorbers on different lines of sight contain information about the degree of clustering. We develop an adaptation of the rigid motion corrected estimator of the reduced second-moment function for the absorber catalog taking into account the across-line-of-sight information. We show how to compute this estimator efficiently using the weights for an isotropic estimator proposed in a recent study. We also show how the modified versions of the rigid motion estimators can be obtained. Simulations suggest that using the modified rigid motion correction estimator may reduce standard errors by 5-20% on scales from 50 to 250 $h^{-1}$ Mpc for a set of 100 lines of sight.
    Full-text · Article · Feb 2003 · Journal of the American Statistical Association
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