Tests of statistical methods for estimating galaxy luminosity function and applications to the Hubble Deep Field: Implication to the cosmic star formation

The luminosity function (LF) of galaxies plays an important role in galactic astronomy and observational cosmology. However, estimation of the LF from observational data is not a trivial task, because of the Aux-limited nature of the astronomical data. Willmer (1997) made an elaborate intercomparison of the estimators by simulations, and their properties have clarified, but it gave rise to some new questions. In this study, we focused on four nonparametric estimators (Schmidt's 1/V-max estimator and three maximum-likelihood estimators). We improved some of these estimators for studies of the distant universe, and examined their performances for much wider class of functional forms by numerical simulation. We found that the Schmidt's 1/V-max estimator yields unbiased result if there is no clustering, but is not robust against clusters or voids. This is consistent with the well-known results. But we did not reconfirm the bias trend of 1/V-max estimator suggested by Willmer (1997). We also found that, in practice, maximum likelihood type estimators are quite robust and give consistent results with each other. Nowadays redshifts of very distant galaxies have been available with the aid of large telescope facilities and photometric redshift techniques, and coherent understanding of the evolution of the LF from the local to distant universe is desired. We apply the methods to the photometric redshift catalog of galaxies in the Hubble Deep Field (HDF) to study the evolution of the LF at very large redshift. At present, the intermediate-high redshift results are yet controversial with each other. We analyzed the new catalog prepared by Fernandez-Soto et al. (1999), and found a rather mild evolution of the LF, consistent with Sawicki et al. (1997).