Extracting a less model dependent cosmic ray composition from $X_\mathrm{max}$ distributions

At higher energies the uncertainty in the estimated cosmic ray mass composition, extracted from the observed distributions of the depth of shower maximum $X_\text{max}$, is dominated by uncertainties in the hadronic interaction models. Thus, the estimated composition depends strongly on the particular model used for its interpretation. To reduce this model dependency in the interpretation of the mass composition, we have developed a novel approach which allows the adjustment of the normalisation levels of the proton $<X_\text{max}>$ and $\sigma (X_\text{max})$ guided by real observations of $X_\text{max}$ distributions. In this paper we describe the details of this approach and present a study of its performance and its limitations. Using this approach we extracted cosmic ray mass composition information from the published Pierre Auger $X_\text{max}$ distributions. We have obtained a consistent mass composition interpretation for Epos-LHC, QGSJetII-04 and Sibyll2.3. Our fits suggest a composition consisting of predominantly iron. Below $10^{18.8}$ eV, the small proportions of proton, helium and nitrogen vary. Above $10^{18.8}$ eV, there is little proton or helium, and with increasing energy the nitrogen component gradually gives way to the growing iron component, which dominates at the highest energies. The fits suggest that the normalisation level for proton $<X_\mathrm{max}>$ is much deeper than the initial predictions of the hadronic interaction models. The fitted normalisation level for proton $\sigma (X_\text{max})$ is also greater than the model predictions. When fixing the expected normalisation of $\sigma(X_\mathrm{max})$ to that suggested by the QGSJetII-04 model, a slightly larger fraction of protons is obtained. These results remain sensitive to the other model parameters that we keep fixed, such as the elongation rate and the $<X_\text{max}>$ separation between p and Fe.