Can the Efron-Petrosian Method Recover the Inverse-Square Distance Law for Simulated Radio Pulsar Fluxes?

TL;DR

This study asks whether the Efron-Petrosian (E-P) method can recover the inverse-square law $F \propto D^{-\alpha}$ for radio pulsar flux using a Parkes-mimicking synthetic catalog generated with { t PsrPopPy}. It finds that, under the realistic SNR-based truncation of the catalog, the E-P statistic $\tau$ does not consistently recover $\alpha=2$, due to a nonlinear, scatter-rich relationship between flux and $\text{SNR}$ that biases the analysis; removing the SNR cutoff restores the expected result. By contrast, truncating the sample with a flux-based cutoff allows robust recovery of the true distance exponent, with pristine agreement for lower flux cuts and within $1\sigma$ for higher cuts, indicating the critical role of survey selection in E-P analyses. The results highlight the need to match truncation criteria to the underlying data-generating process when applying nonparametric methods to astrophysical flux-distance relationships, and demonstrate that flux-based cuts can yield reliable inferences about the distance dependence of pulsar flux.

Abstract

We test whether the Efron-Petrosian (E-P) method can recover the inverse-square law dependence of the radio pulsar flux, using a synthetic catalog generated according to the specifications of the Parkes multi-beam survey using the {\tt PsrPopPy} software. We find that the E-P method cannot reproduce the inverse-square law, except over a narrow range of flux thresholds and even here we don't get pristine agreement. The main reason for the deviation is that the synthetic radio pulsar catalog is truncated based on a cut on the pulsar signal to noise ratio (SNR), which has a non-linear dependence on the flux along with plenty of scatter. We show that the disagreement is exacerbated as we raise the SNR threshold. We then demonstrate that if we create a synthetic catalog based on a flux cut (instead of an SNR-based threshold), we can recover the true distance exponent, with an accuracy ranging from pristine agreement to within $\pm 1 σ$ depending on the chosen flux threshold.

Figures (8)

• Figure 1: Histogram of the logarithm of radio fluxes of pulsars at 1400 MHz for lfl06 radial distribution generated with psrpoppy. The dashed lines $a$, $b$, $c$, $d$ denote the flux thresholds. The thresholds $a$, $b$, $c$, $d$ have been chosen in such a way that $a$ discards $10\%$, $b$ discards $20\%$, $c$ discards $30\%$, and d discards $40\%$ of the pulsars.
• Figure 2: The Efron-Petrosian statistic $\tau$ versus $\alpha$ for different flux thresholds (cf. Fig. \ref{['fig:histo']} computed on the synthetic pulsar dataset generated using PsrPopPy with the lfl06 radial distribution model. The grey shaded region shows the $\pm 1$ range for $\tau$.
• Figure 3: E-P $\tau$ as a function of the flux threshold plot for a fixed distance exponent of 2. The grey shaded region shows the $\pm 1$ range for $\tau$.
• Figure 4: The Efron-Petrosian statistic $\tau$ versus $\alpha$ for different flux thresholds computed on the synthetic pulsar dataset for a SNR cutoff of zero. The grey shaded region shows the $\pm 1$ range for $\tau$. We find pristine agreement with the inverse-square law for all the four flux thresholds.
• Figure 5: Scatter contour plot showing SNR as a function of Flux at 1400 MHz for synthetic pulsars simulated using PsrPopPy. The best-fit regression relation is indicated by the solid red line. We find that there is a non-linear relationship between SNR and flux, along with considerable scatter. Note that that the high density regions have been replaced by contours following the prescription in astroML.