A likelihood method to cross-calibrate air-shower detectors

TL;DR

This paper introduces a maximum-likelihood method for unbiased energy calibration of hybrid air-shower detectors, addressing biases in traditional methods, with practical approximations validated through simulations.

Contribution

It develops a general maximum-likelihood approach for air-shower detector calibration, applicable beyond the Pierre Auger Observatory, and proposes computationally efficient approximations.

Findings

The likelihood-based calibration reduces bias compared to least-squares methods.

Two approximations provide accurate results with lower computational cost.

Validated methods successfully calibrate Pierre Auger Observatory data.

Abstract

We present a detailed statistical treatment of the energy calibration of hybrid air-shower detectors, which combine a surface detector array and a fluorescence detector, to obtain an unbiased estimate of the calibration curve. The special features of calibration data from air showers prevent unbiased results, if a standard least-squares fit is applied to the problem. We develop a general maximum-likelihood approach, based on the detailed statistical model, to solve the problem. Our approach was developed for the Pierre Auger Observatory, but the applied principles are general and can be transferred to other air-shower experiments, even to the cross-calibration of other observables. Since our general likelihood function is expensive to compute, we derive two approximations with significantly smaller computational cost. In the recent years both have been used to calibrate data of the Pierre Auger Observatory. We demonstrate that these approximations introduce negligible bias when they are applied to simulated toy experiments, which mimic realistic experimental conditions.