We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces H _p ^a for p > 0, which are known to be equal to L _p for p > 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of n-term approximation in which a weight function is used to control the terms of the approximations.