Random lasers (RLs) are a special type of laser with a feedback mechanism arising from the random photon scattering in a disordered medium. Their emitted intensity is inherently stochastic. Here we compare results for the intensity distribution from two classes of models. The first concerns electromagnetic wave scattering in a random medium with field amplitudes and phases as independent random or locally correlated variables [random phase sum (RPS)-based models]. In the second, stochastic differential equations describe the mode dynamics in a random medium. Whereas RPS-based models imply Rayleigh, exponential, and K distributions, in the second class we extend to any degree f of optical nonlinearity previous results valid only up to the sixth order, introducing a novel family of intensity distributions, the generalized Izrailev distributions of order f . Model predictions are compared to very large experimental datasets from two quite distinct RLs: a N d 3 + -doped nanopowder and a mixture of colloids containing T i O 2 particles and a dye solution. While RPS models do not provide good data fits, excellent agreement is found with the stochastic differential model, indicating that it properly captures the influence of high-order nonlinearities on the intensity distribution of RLs.