A New Class of Estimators for the [ITAL]N[/ITAL]-Point Correlations

Abstract

A class of improved estimators is proposed for N-point correlation functions of galaxy clustering and for discrete spatial random processes in general. In the limit of weak clustering, the variance of the unbiased estimator converges to the continuum value much faster than with any alternative, and all terms giving rise to a slower convergence exactly cancel. Explicit variance formulae are provided for both Poisson and multinomial point processes using techniques for spatial statistics reported by Ripley. The formalism naturally includes most previously used statistical tools such as N-point correlation functions and their Fourier counterparts, moments of counts in cells, and moment correlators. For all these, and perhaps some other statistics, our estimator provides a straightforward means for efficient edge corrections.