The geometry of second-order statistics - biases in common estimators

Second-order measures, such as the two-point cor- relation function, are geometrical quantities describing the clus- tering properties of a point distribution. In this article well- known estimators for the correlation integral are reviewed and their relation to geometrical estimators for the two-point corre- lation function is put forward. Simulations illustrate the range of applicability of these estimators. The interpretation of the two-point correlation function as the excess of clustering with respect to Poisson distributed points has led to biases in com- mon estimators. Comparing with the approximately unbiased geometrical estimators, we show how biases enter the estima- tors introduced by Davis & Peebles (1983), Landy & Szalay (1993), and Hamilton (1993). We give recommendations for the application of the estimators, including details of the numerical implementation. The properties of the estimators of the corre- lation integral are illustrated in an application to a sample of IRAS galaxies. It is found that, due to the limitations of current galaxy catalogues in number and depth, no reliable determina- tion of the correlation integral on large scales is possible. In the sample of IRAS galaxies considered, several estimators using different finite-size corrections yield different results on scales 1 larger than 20h 1 Mpc, while all of them agree on smaller scales.