  We focus on the primary composition of cosmic rays with the highest energies
that cause extensive air showers in the Earth's atmosphere. A way of examining
the two lowest order moments of the sample distribution of the depth of shower
maximum is presented. The aim is to show that useful information about the
composition of the primary beam can be inferred with limited knowledge we have
about processes underlying these observations. In order to describe how the
moments of the depth of shower maximum depend on the type of primary particles
and their energies, we utilize a superposition model. Using the principle of
maximum entropy, we are able to determine what trends in the primary
composition are consistent with the input data, while relying on a limited
amount of information from shower physics. Some capabilities and limitations of
the proposed method are discussed. In order to achieve a realistic description
of the primary mass composition, we pay special attention to the choice of the
parameters of the superposition model. We present two examples that demonstrate
what consequences can be drawn for energy dependent changes in the primary
composition.
