Papers › A flexible method for estimating luminosity functions via Kernel Density Estimation --...
A flexible method for estimating luminosity functions via Kernel Density Estimation -- II. Generalization and Python implementation
Zunli Yuan, Xibin Zhang, Jiancheng Wang, Xiangming Cheng, Wenjie Wang
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We propose a generalization of our previous KDE (kernel density estimation) method for estimating luminosity functions (LFs). This new upgrade further extend the application scope of our KDE method, making it a very flexible approach which is suitable to deal with most of 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 {Z₁<z<Z₂, L>fₗᵢₘ(z) }. We use the transformation-reflection KDE method (ϕ̂) to solve the problem, and introduce an approximate method (ϕ̂₁) 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 ϕ̂_(bin). Moreover, with the increase of sample size n, our LF estimator converges to the true LF remarkably faster than ϕ̂_(bin). To implement our method, we have developed a public, open-source Python Toolkit, called \texttt{kdeLF}. With the support of \texttt{kdeLF}, our KDE method is expected to be a competitive alternative to existing nonparametric estimators, due to its high accuracy and excellent stability. \texttt{kdeLF} is available at \url{http://github.com/yuanzunli/kdeLF} with extensive documentation available at \url{http://kdelf.readthedocs.org/en/latest~}.
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