A Flexible Method for Estimating Luminosity Functions via Kernel Density Estimation. II. Generalization and Python Implementation

Abstract

Abstract We propose a generalization of our previous kernel density estimation (KDE) method for estimating luminosity functions (LFs). This new upgrade further extends the application scope of our KDE method, making it a very flexible approach that is suitable to deal with most bivariate LF calculation problems. From the mathematical point of view, usually the LF calculation can be abstracted as a density estimation problem in the bounded domain of { Z 1 < z < Z 2 , L > f lim ( z ) } . We use the transformation-reflection KDE method ( ϕ ˆ ) to solve the problem, and introduce an approximate method ( ϕ ˆ 1 ) based on one-dimensional KDE to deal with the small sample size case. In practical applications, the different versions of LF estimators can be flexibly chosen according to the Kolmogorov–Smirnov test criterion. Based on 200 simulated samples, we find that for both cases of dividing or not dividing redshift bins, especially for the latter, our method performs significantly better than the traditional binning method ϕ ˆ bin . Moreover, with the increase of sample size n, our LF estimator converges to the true LF remarkably faster than ϕ ˆ bin . To implement our method, we have developed a public, open-source Python toolkit, called kdeLF. With the support of kdeLF, our KDE method is expected to be a competitive alternative to existing nonparametric estimators, due to its high accuracy and excellent stability. kdeLF is available online at GitHub with further extensive documentation available.