A Statistical Method for Estimating Luminosity Functions using Truncated Data

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 φ(z, M) via log φ(z, M) = f(z) + g(M) + h(z, M, θ) where f and g are estimated nonparametrically, and h takes an assumed parametric form. 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. Subject headings: truncation bias, luminosity function, statistical procedures, quasars