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

TL;DR

This paper clarifies the correct asymptotic distribution of the likelihood ratio test statistic in two-peak experiments with boundary parameters, showing that standard chi-squared assumptions can be misleading and proposing a mixture model for accurate significance estimation.

Contribution

It provides a boundary-aware asymptotic distribution for the likelihood ratio test in two-parameter boundary cases, improving significance calibration in high-energy physics experiments.

Findings

Standard chi-squared assumptions can miscalibrate significance.

The test statistic follows a chi-squared mixture distribution.

Boundary-aware asymptotics improve significance estimation.

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.