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 xi and zeta. 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 sigma(DD)(ens) = [[DD(r)] + 4[DD(r)]2/N(g)]1/2 and sigma(DDD)(ens) = [[DDD(r, u, v)] + 36[DDD(r, u, v)]2/N(g),]1/2 (where N(g) 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 sigma(DD)(bo) = [3DD(r) + 4DD2(r)/N(g)]1/2 and sigma(DDD)(bo) = [7DDD(r, u, v) + 36DDD2(r, u, v)/N(g)]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 xi and the constant Q of zeta) in the regression models. The interdependence among the counts in different bins reduces the fit errors. If we adopt the ensemble errors [sigma(DD)(ens) and sigma(DDD)(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 sigma(DD) and sigma(DDD) 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.