Parameter estimation of a power spectrum equal to a power law + white noise

Question

Given $X_t$ a multivariate random gaussian variable of covariance matrix $N_{tt'}$ diagonal in Fourier space (sampling is equally spaced),

I would like to parametrise its power spectrum as: $S_X(f) = \frac{a}{f^\alpha} + b$, for $f\in[f_{min},f_{max}]$ and estimate $a$, $b$ and $\alpha$.

Any idea how to get these estimators (unbiased if possible), the goal being to use these parameters (and extrapolate to all f) to describe the covariance matrix?

There are plenty of references on how to estimate the slope of a power-law distribution (see review by Clauset et al. 2007, SIAM Proceedings), but in the case I am interested in, the starting point is $X_t$, not $S_X(f)$

Answer 1

1. do a spectrum estimate(How? depends on your process, you might need some multi tapering method.) and fit your function to it.
2. use wavelet, calculate wavelet variance, which is like spectrum but integrated over a band, then fit the integrated line to it.
3. Similar to wavelet, but make use of the fact that wavelet approximated de-correlats, different frequency band for stationary process, figure out the approximated likelihood function to your process by assuming Gaussian, and do a maximum likelihood estimation.

All of these can be found in the book Wavelet Methods for Time Series Analysis by Percival and Walden.