A flexible method for estimating luminosity functions via Kernel Density Estimation - III. Extending to Multiple Flux-Limited Samples

Abstract

As the third paper in a series regarding the estimation of luminosity functions (LFs) via kernel density estimation (KDE), we present a further generalization of our framework by extending its applicability to multiple flux-limited samples. While our previous works addressed single flux-limited datasets, many practical surveys collect data from several disjoint sky regions with varying observational depths and flux limits. We introduce a piecewise estimation framework that partitions the luminosity-redshift plane into disjoint regions according to the staggered flux limits of the sub-samples. Within each region, we integrate data from all surveys capable of detecting sources into a combined sample and apply the transformation-reflection KDE method using the corresponding local flux threshold as the truncation boundary. This strategy allows for the full utilization of all available sources while maintaining rigorous statistical consistency. The robustness of this approach is validated through Monte Carlo simulations. Furthermore, application to SDSS DR7 and 2SLAQ quasar data shows smooth transitions across flux limits and excellent agreement with parametric models. This approach is fully supported by our previously developed \texttt{kdeLF} Python package.

Figures (12)

• Figure 1: Schematic illustration of the domain partition for the two-sample case. Sample A represents the deeper survey with a lower flux limit $f_{\mathrm{lim}}^{(1)}(z)$, while Sample B is the shallower survey with a limit $f_{\mathrm{lim}}^{(2)}(z)$. The flux curves divide the accessible $L$--$z$ plane into two disjoint regions: the faint interval ($f_{\mathrm{lim}}^{(1)} < L \le f_{\mathrm{lim}}^{(2)}$), which is populated exclusively by sources from Sample A, and the bright interval ($L > f_{\mathrm{lim}}^{(2)}$), where sources are detectable by both surveys. This segmentation forms the basis for our piecewise LF estimation.
• Figure 2: Median LFs and uncertainties derived from 200 simulated realizations using our generalized KDE method (specifically the adaptive estimator, $\hat{\phi}_{\mathrm{a}}$). The panels show the results at four representative redshifts: $z=0.5, 1.0, 2.0$, and $3.5$. The red solid curves represent the median of the 200 estimates, while the orange shaded regions indicate the $1\,\sigma$ dispersion across the 200 samples. The green dashed curves show the ground-truth input LF (Model A from 2017ApJ...846...78Y). The vertical gray lines (dotted, dashed-dotted, and dashed) mark the positions of the flux limits $f_{\mathrm{lim}}^{(1)}$, $f_{\mathrm{lim}}^{(2)}$, and $f_{\mathrm{lim}}^{(3)}$, respectively, corresponding to the three nested survey tiers.
• Figure 3: Distribution of the quasar sample compiled by 2019MNRAS.488.1035K in the $M_{1450}$--$z$ plane, restricted to $1.0 \le z < 2.2$. Red points show the SDSS DR7 subsample (32,548 quasars; wide area, relatively shallow), while green points indicate the 2SLAQ subsample (7,090 quasars; narrower area, deeper). The dashed lines indicate the respective flux-limit curves for the two surveys. This visualization highlights the tiered coverage of the datasets, where the two samples overlap in $M_{1450}$ while sampling different comoving volumes.
• Figure 4: Quasar UV LF estimates at four representative redshifts within $1.0 < z < 2.2$. The red solid curves represent our fiducial generalized KDE estimates (using the $\hat{\phi}_{\mathrm{a}}$ estimator), with orange shaded regions indicating the $3\sigma$ ($99.93\%$) uncertainties derived from MCMC analysis. The green dashed lines denote the reference LF from 2019MNRAS.488.1035K. Vertical grey dash-dotted and dashed lines mark the luminosity limits of the 2SLAQ and SDSS DR7 surveys, respectively, highlighting the piecewise boundaries of the tiered observational data.
• Figure 5: Posterior distributions of the adaptive KDE parameters derived using the kdeLF package. Left and right panels correspond to the first and second steps of our piecewise estimation, respectively.