Estimating third-order moments for an absorber catalog

Thanks to the recent availability of large surveys, there has been renewed interest in third-order correlation statistics. Measures of third-order clustering are sensitive to the structure of filaments and voids in the universe and are useful for studying large-scale structure. Thus, statistics of these third-order measures can be used to test and constrain parameters in cosmological models. Third-order measures such as the three-point correlation function are now commonly estimated for galaxy surveys. Studies of third-order clustering of absorption systems will complement these analyses. We define a statistic, which we denote K, that measures third-order clustering of a data set of point observations and focus on estimating this statistic for an absorber catalog. The statistic K can be considered a third-order version of the second-order Ripley K-function and allows one to study the abundance of various configurations of point triplets. In particular, configurations consisting of point triplets that lie close to a straight line can be examined. Studying third-order clustering of absorbers requires consideration of the absorbers as a three-dimensional process, observed on QSO lines of sight that extend radially in three-dimensional space from Earth. Since most of this three-dimensional space is not probed by the lines of sight, edge corrections become important. We use an analytical form of edge correction weights and construct an estimator of the statistic K. for use with an absorber catalog. We show that with these weights, ratio-unbiased estimates of . can be obtained. Results from a simulation study also verify unbiasedness and provide information on the decrease of standard errors with increasing number of lines of sight.