A Flexible Method for Estimating Luminosity Functions via Kernel Density Estimation

Abstract

We propose a flexible method for estimating luminosity functions (LFs) based on kernel density estimation (KDE), the most popular nonparametric density estimation approach developed in modern statistics, to overcome issues surrounding binning of LFs. One challenge in applying KDE to LFs is how to treat the boundary bias problem, since astronomical surveys usually obtain truncated samples predominantly due to the flux-density limits of surveys. We use two solutions, the transformation KDE method (φ t ), and the transformation-reflection KDE method (φ tr ) to reduce the boundary bias. We develop a new likelihood cross-validation criterion for selecting optimal bandwidths, based on which, the posterior probability distribution of bandwidth and transformation parameters forφ t andφ tr are derived within a Markov chain Monte Carlo (MCMC) sampling procedure. The simulation result shows thatφ t andφ tr perform better than the traditional binned method, especially in the sparse data regime around the flux-limit of a survey or at the bright-end of the LF. To further improve the performance of our KDE methods, we develop the transformation-reflection adaptive KDE approach (φ tra ). Monte Carlo simulations suggest that it has a good stability and reliability in performance, and is around an order of magnitude more accurate than using the binned method. By applying our adaptive KDE method to a quasar sample, we find that it achieves estimates comparable to the rigorous determination in a previous work, while making far fewer assumptions about the LF. The KDE method we develop has the advantages of both parametric and non-parametric methods.

On the Cosmic Evolution of AGN Obscuration and the X-Ray Luminosity Function: XMM-Newton and Chandra Spectral Analysis of the 31.3 deg2 Stripe 82X

“…Even if they both affect the detection of sources with high N H , the sky coverage bias compensates for the fact that different areas are sensitive to different observed fluxes, while the Malmquist bias depends on the intrinsic properties (z, L X , and N H ) of the sources. We corrected for the sky coverage bias by weighting each source by the reciprocal of its sky coverage at the corresponding observed flux (e.g., Liu et al 2017;Yuan et al 2020). The weights were then included in the simulations performed in Section 5.1.1.…”

“…The derived binned LF is consistent, within the uncertainties, with the XLFs from U14, A15, and A19. To fit the XLF, we applied the kernel density estimation (KDE) method described by Yuan et al (2020Yuan et al ( , 2022. This nonparametric approach takes advantage of the mathematics beyond the KDE (Wasserman 2006), a wellestablished procedure to estimate continuous density functions (e.g., Chen 2017; Davies & Baddeley 2018).…”

“…The derived XLF is shown in Figure 15 (solid red lines and orange contours). In principle, the KDE method allows for extrapolation of the XLF beyond the luminosity limits of the observations; however, the resulting high-luminosity tail may overestimate the true XLF, especially when using the adaptive method (Yuan et al 2020), so our XLFs are reported only for observed luminosities. As shown by the posterior error estimations in Figure 16, the derived best-fit parameters are well constrained, meaning a good kernel estimation (Y22).…”

The XXL survey

“…This expression is the one found commonly in the literature (e.g. Kelly et al 2008;Yuan et al 2020). Furthermore, this expression can be generalised naturally to multiple fields j with different detection limits and observational areas as:…”

Two-point statistics without bins: A continuous-function generalization of the correlation function estimator for large-scale structure

“…This is useful when we want to reconstruct a distribution without making many assumptions about the data, as is required in parametric methods. KDEs have found use in many areas of astrophyics, for example to measure the 21cm power spectrum with reduced foreground contamination (Trott et al 2019), and to estimate luminosity functions with superior performance compared to binned methods (Yuan et al 2020). Hatfield et al (2016) uses a KDE approach to estimate the angular correlation function, in order to address the issues of information loss and arbitrary bin choice inherent to binning; they optimize for the kernel choice, and find a correlation function consistant with that of the binned method.…”