## The geometry of second-order statistics - biases in common estimators
Second-order measures, such as the two-point correlation function,
are geometrical quantities describing the clustering 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 correlation 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 common estimators. Comparing with the approximately unbiased
geometrical estimators, we show how biases enter the estimators
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 correlation 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 determination 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
This article contains no SIMBAD objects. ## Contents- 1. Introduction
- 2. Estimators for the correlation integral
*C*(*r*) - 3. Geometrical estimators for the two-point correlation function
*g*(*r*) - 4. Estimators for the two-point correlation function
*g*(*r*) based on*DR*and*RR* - 5. Improved estimators for
*C*(*r*) and*g*(*r*) - 6. Correlation integral of IRAS galaxies
- 6.1. A note on scaling
- 7. Remarks
- 8. Conclusions and recommended estimators
- Acknowledgements
- Appendix
- References
© European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |