A Maximum Likelihood Approach to Estimating Correlation Functions

TL;DR

This paper introduces a maximum likelihood estimator for the correlation function that achieves higher precision with fewer random points, improving measurement accuracy without significant additional computational cost.

Contribution

The paper develops a maximum likelihood estimator for the correlation function that outperforms the Landy and Szalay estimator in accuracy and efficiency.

Findings

ML estimator yields smaller measurement errors than LS estimator.

ML estimator requires fewer random points for the same precision.

Implementation of ML estimator involves minimal code changes.

Abstract

We define a Maximum Likelihood (ML for short) estimator for the correlation function, {\xi}, that uses the same pair counting observables (D, R, DD, DR, RR) as the standard Landy and Szalay (1993, LS for short) estimator. The ML estimator outperforms the LS estimator in that it results in smaller measurement errors at any fixed random point density. Put another way, the ML estimator can reach the same precision as the LS estimator with a significantly smaller random point catalog. Moreover, these gains are achieved without significantly increasing the computational requirements for estimating {\xi}. We quantify the relative improvement of the ML estimator over the LS estimator, and discuss the regimes under which these improvements are most significant. We present a short guide on how to implement the ML estimator, and emphasize that the code alterations required to switch from a LS to a ML estimator are minimal.