On the error estimates of correlation functions

In this paper, we derive analytical formulae for ensemble and bootstrap-resampling errors in two- and three-point correlation functions ξ and ζ. The analytical results agree well with numerical simulations. The ensemble errors of pair count DD (r) and triplet count DDD (r, u, v) are well approximated by σDD_ (ens) =+4²^/Ng_¹/2^ and σDDD_ (ens) = +36 (DDD (r, u, v) >²^/Ng_¹/2^ (where Ng_ is the number of points in the sample; denotes ensemble mean), respectively. The bootstrap-resampling errors of DD (r) and DDD (r, u, v) can be approximated by σDD_ (b0) = 3DD (r) +4DD²^ (r) /Ng_¹/2^ and σDDD_ (b0) = 7DDD (r, u, v) +36DDD²^ (r, u, v) /Ng_¹/2^, respectively. Similar derivations are carried out for sparse-sampling errors. We also discuss the fit errors of the parameters (i. e. , the amplitude A of ξ and the constant Q of ζ) in the regression models. The interdependence among the counts in different bins reduces the fit errors. If we adopt the ensemble errors σDD_ (ens) and σDDD_ (ens) for the counts, the fit errors of A and Q in each sample are about half of the standard errors obtained from the ensemble of samples. The underestimation of the fit errors due to the bin-bin interdependence is compensated by the overestimation of σDD_ and σDDD_ given by the bootstrap-resampling method. The fit errors of the parameters, given by the bootstrap-resampling errors for the counts, give the correct answers.