A new method in the microscopic theory of anharmonic crystal lattice dynamics - the independent anharmonic oscillator (IAO) approximation - is discussed. The method is compared with traditional approaches (such as the mean-field and renormalized self-consistent phonon methods) from the standpoint of the analysis of the main approximations. Different methods are applied to the description of the structural phase transition in a monoatomic anharmonic crystal, the results obtained are analyzed, and the numerical accuracy of different approximations is estimated. It is demonstrated that, for this model in the displacement-type instability range, the new method, unlike the self-consistent phonon method, adequately describes the phase transition as a second-order transition. The phase transition temperature calculated in the displacement limit monotonically tends to the exact value in contrast with the temperature obtained within the mean field approximation.