An Improved Method of Constructing Binned Luminosity Functions

We show that binned differential luminosity functions constructed using the 1/Va method have a significant systematic error for objects close to their parent sample's flux limit(s). This is particularly noticeable when luminosity functions are produced for a number of different redshift ranges as is common in the study of AGN or galaxy evolution. We present a simple method of constructing a binned luminosity function which overcomes this problem and has a number of other advantages over the traditional 1/Va method. We also describe a practical method for comparing binned and model luminosity functions, by calculating the expectation values of the binned luminosity function from the model. Binned luminosity functions produced by the two methods are compared for simulated data and for the Large Bright QSO Survey (LBQS). It is shown that the 1/Va method produces a very misleading picture of evolution in the LBQS. The binned luminosity function of the LBQS is then compared to a model two power law luminosity function undergoing pure luminosity evolution from Boyle et al. (1991). The comparison is made using a model luminosity function averaged over each redshift shell, and using the expectation values for the binned luminosity function calculated from the model. The luminosity function averaged in each redshift shell gives a misleading impression that the model over predicts the number of QSOs at low luminosity even when model and data are consistent. The expectation values show that there are significant differences between model and data: the model overpredicts the number of low luminosity sources at both low and high redshift. The luminosity function does not appear to steepen relative to the model as redshift increases.