Over the last two decades, it has been argued that the Lorentz transformation mechanism, which imposes the generalization of Newton's classical mechanics into Einstein's special relativity, implies a generalization, or deformation, of the ordinary statistical mechanics. The exponential function, which defines the Boltzmann's factor, emerges properly deformed within this formalism. Starting from this, so-called $κ$-deformed exponential function, we introduce new classes of statistical distributions emerging as the $κ$-deformed version of already known distribution as the Generalized Gamma, Weibull, Logistic which can be adopted in the analysis of statistical data that exhibit power-law tails.