In this paper, we focus on the problem of speckle removal by means of anisotropic diffusion and, specifically, on the importance of the correct estimation of the statistics involved. First, we derive an anisotropic diffusion filter that does not depend on a linear approximation of the speckle model assumed, which is the case of a previously reported filter, namely, SRAD. Then, we focus on the problem of estimation of the coefficient of variation of both signal and noise and of noise itself. Our experiments indicate that neighborhoods used for parameter estimation do not need to coincide with those used in the diffusion equations. Then, we show that, as long as the estimates are good enough, the filter proposed here and the SRAD perform fairly closely, a fact that emphasizes the importance of the correct estimation of the coefficients of variation.