A method for determining luminosity functions incorporating both flux measurements and flux upper limits, with applications to the average X-ray to optical luminosity ratio for quasars.

A novel method for determining luminosity functions from preselected random samples is derived. The method incorporates positive flux measurements as well as flux upper limits. It is nonparametric, and enables one to determine the best estimate and ranges of uncertainty for the shape of the luminosity function, and for any physical quantity that depends on the shape of the luminosity function. By applying this method to preliminary results from the quasar-survey of the Einstein observatory, we determine the distribution of the values of L_x/L_opt for quasars, and the average of this ratio. The method is very general and can be used to determine any type of distribution function (not only luminosity functions). It can also be formulated as a parametric method, and be used to determine the parameters of distribution functions with presumed functional forms.