Ultra-high energy cosmic rays (UHECRs) interacting with the atmosphere generate extensive air showers (EAS) of secondary particles. The depth corresponding to the maximum development of the shower, $Xmax$, is a well-known observable for determining the nature of the primary cosmic ray which initiated the cascade process. In this paper, we present an empirical model to describe the distribution of $Xmax$ for EAS initiated by nuclei, in the energy range from $10^{17}$ eV up to $10^{21}$ eV, and by photons, in the energy range from $10^{17}$ eV up to $10^{19.6}$ eV. Our model adopts the generalized Gumbel distribution motivated by the relationship between the generalized Gumbel statistics and the distribution of the sum of non-identically distributed variables in dissipative stochastic systems. We provide an analytical expression for describing the $Xmax$ distribution for photons and for nuclei, and for their first two statistical moments, namely $langle Xmaxrangle$ and $sigma^{2}(Xmax)$. The impact of the hadronic interaction model is investigated in detail, even in the case of the most up-to-date models accounting for LHC observations. We also briefly discuss the differences with a more classical approach and an application to the experimental data based on information theory.
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