Likelihood-ratio tests are widely used in high-energy physics to translate a fit into a p-value and a quoted “sigma” significance. Under regular conditions, Wilks’ theorem states that twice the log-likelihood ratio approaches a chi-squared distribution with one degree of freedom for each parameter being tested. That result depends on conditions including identifiability, parameter-independent support, an interior true parameter, smoothness of the log-likelihood and satisfaction of Bartlett’s identities. Some are violated when physical constraints place a parameter at the edge of its permitted range, breaking the symmetry on which the conventional asymptotic result relies.

Signal yields in counting experiments cannot be negative, so a background-only null hypothesis places a tested signal amplitude at zero while allowing estimates only in the positive direction. For one constrained parameter, the established boundary limit is Chernoff’s 50:50 mixture of a point mass at zero and a chi-squared distribution with one degree of freedom. The new analysis shows that this one-parameter shortcut does not extend to a two-peak alternative. When two expected signal yields, μ1 and μ2, are both treated as parameters of interest and both are fixed at zero under the null, the asymptotic law is a three-component mixture of chi-squared distributions with zero, one and two degrees of freedom.

The mixture weights are determined by the correlation structure of the two signal estimates as encoded in the Fisher information matrix at the null. For two parameters of interest, the weight on the two-degree-of-freedom component is the arccosine of the normalized off-diagonal information divided by two times pi; the remaining weight is assigned to the zero-degree-of-freedom component, while the one-degree-of-freedom component always has weight one-half. The researchers derive analytic information matrices for their double-peak model and compare the resulting references with 100,000 simulated background-only data sets. The findings are not a universal replacement for every counting analysis, but a demonstration that the structure and parametrization of the boundary problem determine the appropriate asymptotic distribution.

An important technical issue is how independent information accumulates. The extended likelihood used for Poisson point-process data combines the probability of the total event count with the conditional distribution of the observed energies, and it does not factor into independent contributions from individual observed events alone. The natural observational units are therefore the complete point pattern or independent exposure blocks, even though the common notation leaves this structure implicit. If only the expected background count is allowed to grow while the signal count is held fixed, the Fisher information for the signal amplitudes tends to zero instead of increasing with the data. That behavior is of limited consequence for the one-parameter boundary limit, whose weights are fixed, but it is critical for the two-parameter mixture because its weights depend directly on the information matrix.

To represent the intended asymptotic sequence, the study introduces an explicit observation duration T and parameterizes expected signal and background yields as rates per unit exposure, μj = T μ̃j and β = T β̃, with the rate parameters held fixed as T increases. In this formulation, signal-related Fisher information grows linearly with exposure. The simulations demonstrate that this is not a cosmetic change. With the two signal amplitudes treated as parameters of interest and the background parameters fixed, Wilks’ regular χ²₂ reference was too low near the origin and too light-tailed, while Chernoff’s one-parameter mixture performed poorly because it represented only one constrained amplitude. A two-parameter mixture derived under a fixed-count parametrization improved the comparison but did not track the empirical distribution across the full range. The exposure-based rate mixture followed the simulated null distribution most closely.

Estimating the background normalization and shape as nuisance parameters changed the calculation again. The researchers partition the parameter vector into the two signal rates and the background-related nuisance parameters, then compute the effective information for the signal rates through the Schur complement. This correction removes information shared between the signal and nuisance blocks. A mixture based on the unadjusted signal-information block no longer agreed with the simulations, whereas the Schur-complement calculation restored close agreement only in the exposure-based rate parametrization. The results therefore show that applying a two-boundary mixture is not sufficient: the information used to calculate its weights must describe both the actual asymptotic sequence and the effect of profiling over nuisance parameters.

The study also examined the common practice of testing one peak while treating the other peak’s non-negative yield as a nuisance parameter, as can arise in searches for individual resonances or lines. When background parameters were fixed, the two signal-rate estimators were negatively correlated, with a correlation of −0.37 in the example, because an observed excess could be attributed to either peak. In this negative-correlation regime, the limiting distribution is not exactly a chi-squared mixture, and the simulations agreed best with an available modified-mixture approximation. Chernoff’s one-boundary distribution was inadequate because a nuisance amplitude constrained to be non-negative still lies on the parameter-space boundary, even though it is not the primary parameter of interest.

Profiling the background parameters as well as the second peak reversed the estimated correlation to positive, reaching 0.55 in the toy example. The two signal rates then tended to move together rather than compete for the same excess, placing the analysis in the positive-correlation regime where a chi-squared mixture with weights determined by the effective Fisher information applies. The empirical null distribution closely followed that prediction, while the approximation designed for the negative-correlation case visibly failed. This reversal shows that adding a small set of nuisance parameters can change not only the precision of a likelihood fit but the qualitative form of its asymptotic reference distribution, making an explicit check of the effective information essential.

These differences propagate directly into discovery thresholds. In the simulations, the Wilks χ²₂ law supplied an overly heavy-tailed reference and therefore conservative significances, potentially missing genuine signals. Chernoff’s one-parameter mixture supplied an overly light-tailed reference, producing artificially small p-values, inflated significance and a greater risk of false discovery claims. The correctly specified mixtures shifted the test-statistic thresholds associated with nominal three-, four- and five-sigma levels. The results have direct implications for double-line gamma-ray searches and collider resonance spectroscopy, and the same boundary-aware framework can be considered for axion-line searches, paired kinematic features and combined signals across channels when several peak amplitudes must remain non-negative. The approximation is asymptotic rather than exact: in the settings examined, deviations began to appear at expected counts around 20, so its accuracy in smaller samples is limited.

Subject of Research: Boundary-aware likelihood-ratio asymptotics for two-peak discovery experiments

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

Article References: Bertinelli Salucci, C., Reiersrud, H. B., Read, A. L., Kvellestad, A., & De Bin, R. (2026). The asymptotic distribution of the likelihood ratio test statistic in two-peak discovery experiments. The European Physical Journal C, 86(10), Article 1198. https://doi.org/10.1140/epjc/s10052-026-16439-9

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16439-9

Keywords: likelihood ratio test, two-peak search, chi-squared mixture, Fisher information, high-energy physics, nuisance parameters, discovery significance, Monte Carlo simulation, asymptotic, distribution, likelihood, ratio