The method of correlation function has been utilized to discuss the nonlinear effects in the ferromagnetic resonance at high power levels. For ordinary ferromagnetic crystals, the present theory includes the results of the semiclassical theory developed by H. Suhl as a classical limit. In the case of YIG doped with rare-earth ions, the frequency shift and the additional losses due to the rare-earth ions have been discussed. Besides, the threshold field of the so-called first order process with parallel pumping and the first and second order processes with perpendicular pumping have also been derived. The results obtained show that the rf critical magnetic field increases with the addition of rare-earth ions. In samples with small concentration of rare-earth ions, the influence on the threshold is attributed to the increase of the losses of spin waves. The temperature dependence and the anisotropy of the threshold field are entirely similar to those of the ferromagnetic resonance linewidth at low power level. The experimental results may be satisfactorily explained by the present theory.