  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.
