  The observational limitations of astronomical surveys lead to significant
statistical inference challenges. One such challenge is the estimation of
luminosity functions given redshift $z$ and absolute magnitude $M$ measurements
from an irregularly truncated sample of objects. This is a bivariate density
estimation problem; we develop here a statistically rigorous method which (1)
does not assume a strict parametric form for the bivariate density; (2) does
not assume independence between redshift and absolute magnitude (and hence
allows evolution of the luminosity function with redshift); (3) does not
require dividing the data into arbitrary bins; and (4) naturally incorporates a
varying selection function. We accomplish this by decomposing the bivariate
density into nonparametric and parametric portions. There is a simple way of
estimating the integrated mean squared error of the estimator; smoothing
parameters are selected to minimize this quantity. Results are presented from
the analysis of a sample of quasars.
