This paper provides a theoretical analysis of the properties of Wavelet based maximum likelihood estimation of the parameters describing 1/f processes embedded in white noise. This analysis shows that such a scheme is only consistent for spectral exponents gamma in the range (0,1). This is in contradiction to the results suggested in previous empirical studies. When gamma in (0,1) this paper also establishes that Wavelet based maximum likelihood methods are asymptotically Gaussian and efficient. Finally, the asymptotic rate of mean--square convergence of the parameter estimates is established and is shown to slow as gamma approaches one.