[Paper Review] Implementation of a goodness-of-fit test for finding optimal concurrent radio and γ-ray pulsar light curves

A. S. Seyffert, C. Venter+3 more|arXiv (Cornell University)|Nov 3, 2016

Statistical and numerical algorithms3 citations

TL;DR

This paper proposes a statistically rigorous, multi-wavelength goodness-of-fit test to simultaneously constrain the viewing geometry (inclination α and observer line-of-sight ζ) of radio- and γ-ray pulsars using geometric emission models. By combining χ² statistics from radio and γ-ray light curves with normalized, scaled test statistics and confidence contours derived from χ² distributions, the method produces consistent, objective constraints that align closely with subjective by-eye fitting results, offering a reproducible alternative to manual fitting.

ABSTRACT

Since the launch of the Fermi Large Area Telescope in 2008 the number of known $γ$-ray pulsars has increased immensely to over 200, many of which are also visible in the radio and X-ray bands. Seyffert et al. (2011) demonstrated how constraints on the viewing geometries of some of these pulsars could be obtained by comparing their observed radio and $γ$-ray light curves by eye to light curves from geometric models. While these constraints compare reasonably well with those yielded by more rigorous single-wavelength approaches, they are still a somewhat subjective representation of how well the models reproduce the observed radio and $γ$-ray light curves. Constructing a more rigorous approach is, however, made difficult by the large uncertainties associated with the $γ$-ray light curves as compared to those associated with the radio light curves. Naively applying a $χ^{2}$-like goodness-of-fit test to both bands invariably results in constraints dictated by the radio light curves. A number of approaches have been proposed to address this issue. In this paper we investigate these approaches and evaluate the results they yield. Based on what we learn, we implement our own version of a goodness-of-fit test, which we then use to investigate the behaviour of the geometric models in multi-dimensional phase space.

Motivation & Objective

• To address the challenge of fitting radio and γ-ray pulsar light curves simultaneously despite their vastly different photon fluxes and uncertainties.
• To develop a statistically robust alternative to subjective by-eye fitting methods that currently dominate multiwavelength pulsar modeling.
• To ensure both radio and γ-ray data contribute equally to the goodness-of-fit evaluation, avoiding bias toward the higher-signal radio data.
• To produce confidence contours in the α–ζ parameter space that are reproducible, objective, and comparable to those from manual fitting.
• To enable systematic comparison of different geometric model combinations using a consistent, scalable test statistic.

Proposed method

• The method computes separate χ² test statistics for radio and γ-ray light curves using model-predicted and observed intensities.
• Each individual χ² statistic is normalized and scaled to account for differing signal-to-noise ratios between radio and γ-ray data.
• The combined test statistic is defined as the sum of the normalized, scaled χ² values from both bands.
• The combined map is minimized over all parameters except α and ζ, resulting in a 2D α–ζ map of goodness of fit.
• Confidence contours (1σ, 2σ, 3σ) are derived by adding critical χ² values (with 2 degrees of freedom) to the global minimum of the combined map.
• The method allows for model comparison by preserving relative scale differences in the test statistics, even when normalization is applied.

Experimental results

Research questions

• RQ1 How can a statistically consistent goodness-of-fit test be constructed when fitting radio and γ-ray pulsar light curves with vastly different uncertainties?
• RQ2 Can a combined χ² test be designed to fairly weight both radio and γ-ray data without bias toward the higher-signal radio band?
• RQ3 How should confidence contours in the α–ζ parameter space be defined when combining two χ² statistics with different scaling properties?
• RQ4 To what extent do the results of this method match those obtained via subjective by-eye fitting, which are considered a de facto standard?
• RQ5 Can this method be used to objectively compare multiple geometric model combinations for pulsar emission?

Key findings

• The proposed method produces confidence contours in α–ζ space that are in strong agreement with those derived from subjective by-eye fitting, validating its reliability.
• The inclusion of confidence contours based on χ² distribution quantiles (1σ, 2σ, 3σ) provides a statistically sound framework for uncertainty estimation.
• The method remains robust even when the normalization of individual test statistics is adjusted, allowing for fair comparison between different model combinations.
• The combined test statistic is insensitive to the omission of the normalization step in the radio component, preserving the shape of the map.
• The approach successfully mitigates the bias toward radio data that plagues naive χ² fitting, ensuring balanced contribution from both bands.
• The method enables objective, repeatable, and scalable fitting of multiwavelength pulsar light curves, paving the way for systematic model comparison.