The asymptotic distribution of the likelihood ratio test statistic in two-peak discovery experiments

TL;DR

This paper addresses the breakdown of Wilks' theorem for likelihood ratio tests when two non-negative parameters lie on the boundary in counting experiments with two nearby signal peaks. It shows that the correct asymptotic distribution is a chi-squared mixture with weights determined by the profiled Fisher information, and that an exposure-based Poisson point-process parametrization is essential to restore Bartlett's identities and obtain valid limits. Through extensive Monte Carlo studies, the authors quantify how parametrization, nuisance parameters, and parameter correlation shape the LR distribution, and they provide explicit weight formulas for the two-boundary and mixed cases. The findings have broad practical impact for two-peak discovery tests in high-energy physics (e.g., gamma-ray lines, collider resonances) by enabling boundary-aware p-value calibration and reducing miscalibration risks in discovery claims.

Abstract

Likelihood ratio tests are widely used in high-energy physics, where the test statistic is usually assumed to follow a chi-squared distribution with a number of degrees of freedom specified by Wilks' theorem. This assumption breaks down when parameters such as signal or coupling strengths are restricted to be non-negative and their values under the null hypothesis lie on the boundary of the parameter space. Based on a recent clarification concerning the correct asymptotic distribution of the likelihood ratio test statistic for cases where two of the parameters are on the boundary, we revisit the the question of significance estimation for two-peak signal-plus-background counting experiments. In the high-energy physics literature, such experiments are commonly analyzed using Wilks' chi-squared distribution or the one-parameter Chernoff limit. We demonstrate that these approaches can lead to strongly miscalibrated significances, and that the test statistic distribution is instead well described by a chi-squared mixture with weights determined by the Fisher information matrix. Our results highlight the need for boundary-aware asymptotics in the analysis of two-peak counting experiments.

Figures (6)

• Figure 1: Illustration of the two-peak toy model used throughout the analysis. The green area corresponds to the background component, modelled as a normalized exponential density; the red area represents the sum of two truncated Gaussian signal components (double-peak signal); and the grey histogram shows a realization of data simulated under the null hypothesis.
• Figure 2: Diagnostic for the estimation of the two expected signal counts without nuisance parameters. Left: Empirical cumulative distribution function (cdf) of $\mathop{\mathrm{\lambda_{\hbox{\scriptsize LR}}}}\nolimits$ (solid black), overlaid with four reference cdfs: (i) the Wilks' regular case, a $\chi^2$ with 2 degrees of freedom (orange dashed); (ii) the Chernoff's one-parameter boundary law $\frac{1}{2} \chi_0^2 + \frac{1}{2} \chi_1^2$ (teal dotted); (iii) the chi-squared mixture limit for two parameters on the boundary from SL, under the non-exposure parametrization (purple long-dashed); (iv) the same chi-squared mixture, with the exposure-based parametrization (light-green dash-dot). The inset table reports the $50^{\hbox{\footnotesize th}}$, $95^{\hbox{\footnotesize th}}$, and $99^{\hbox{\footnotesize th}}$ quantiles for each curve, with bold highlighting the closest entries to the empirical quantiles. Right: residual plot $F_{\hbox{\footnotesize ref}} (t) - F_{\hbox{\footnotesize emp}} (t)$, where positive values indicate lighter tail than observed, negative values indicate heavier tail.
• Figure 3: Diagnostic for the estimation of the two expected signal counts and the background nuisance parameters. Left: Empirical cumulative distribution function (cdf) of $\mathop{\mathrm{\lambda_{\hbox{\scriptsize LR}}}}\nolimits$ (solid black), overlaid with six reference cdfs: (i) the Wilks' regular case, a $\chi^2$ with 2 degrees of freedom (orange dashed); (ii) the Chernoff's one-parameter boundary law $\frac{1}{2} \chi_0^2 + \frac{1}{2} \chi_1^2$ (teal dotted); (iii) the chi-squared mixture limit for two parameters on the boundary from SL, under the non-exposure parametrization (purple long-dashed); (iv) the same chi-squared mixture, with the exposure-based parametrization (light-green dash-dot); (v) the exposure-based profile chi-squared mixture (cyan long-dash); (vi) the single-event/no-exposure profile chi-squared mixture (dark-blue dash–dot). The inset table reports the $50^{\hbox{\footnotesize th}}$, $95^{\hbox{\footnotesize th}}$, and $99^{\hbox{\footnotesize th}}$ quantiles for each curve, with bold highlighting the closest entries to the empirical quantiles. Right: residual plot $F_{\hbox{\footnotesize ref}} (t) - F_{\hbox{\footnotesize emp}} (t)$, where positive values indicate lighter tail than observed, negative values indicate heavier tail.
• Figure 4: Diagnostic for the estimation of the first peak yield with the second as nuisance parameter and no additional nuisance from the background. Left: Empirical cumulative distribution function (cdf) of $\mathop{\mathrm{\lambda_{\hbox{\scriptsize LR}}}}\nolimits$ (solid black), overlaid with four reference cdfs: (i) the Wilks' regular case, a $\chi^2$ with 2 degrees of freedom (orange dashed); (ii) the Chernoff's one-parameter boundary law $\frac{1}{2} \chi_0^2 + \frac{1}{2} \chi_1^2$ (teal dotted); (iii) the chi-squared mixture limit for two parameters on the boundary from SL, under the non-exposure parametrization (light-green dash-dot); (iv) the modified distribution for negatively correlated parameters from us (red double-dash). The inset table reports the $50^{\hbox{\footnotesize th}}$, $95^{\hbox{\footnotesize th}}$, and $99^{\hbox{\footnotesize th}}$ quantiles for each curve, with bold highlighting the closest entries to the empirical quantiles. Right: residual plot $F_{\hbox{\footnotesize ref}} (t) - F_{\hbox{\footnotesize emp}} (t)$, where positive values indicate lighter tail than observed, negative values indicate heavier tail.
• Figure 5: Diagnostic for the estimation of the first peak yield with the second as nuisance parameter and the three additional nuisance from the background. Left: Empirical cumulative distribution function (cdf) of $\mathop{\mathrm{\lambda_{\hbox{\scriptsize LR}}}}\nolimits$ (solid black), overlaid with five reference cdfs: (i) the Wilks' regular case, a $\chi^2$ with 2 degrees of freedom (orange dashed); (ii) the Chernoff's one-parameter boundary law $\frac{1}{2} \chi_0^2 + \frac{1}{2} \chi_1^2$ (teal dotted); (iii) the chi-squared mixture limit for two parameters on the boundary from SL, under the non-exposure parametrization (light-green dash-dot); (iv) the modified distribution for negatively correlated parameters from us (red double-dash); (v) the exact chi-squared mixture for the positive correlation case (yellow long-dashed). The inset table reports the $50^{\hbox{\footnotesize th}}$, $95^{\hbox{\footnotesize th}}$, and $99^{\hbox{\footnotesize th}}$ quantiles for each curve, with bold highlighting the closest entries to the empirical quantiles. Right: residual plot $F_{\hbox{\footnotesize ref}} (t) - F_{\hbox{\footnotesize emp}} (t)$, where positive values indicate lighter tail than observed, negative values indicate heavier tail.