  We discuss an approach to the component separation of microwave,
multi-frequency sky maps as those typically produced from Cosmic Microwave
Background (CMB) Anisotropy data sets. The algorithm is based on the two step,
parametric, likelihood-based technique recently elaborated on by Eriksen et
al., (2006), where the foreground spectral parameters are estimated prior to
the actual separation of the components. In contrast with the previous
approaches, we accomplish the former task with help of an analytically-derived
likelihood function for the spectral parameters, which, we show, yields
estimates equal to the maximum likelihood values of the full multi-dimensional
data problem. We then use these estimates to perform the second step via the
standard, generalized-least-square-like procedure. We demonstrate that the
proposed approach is equivalent to a direct maximization of the full data
likelihood, which is recast in a computationally tractable form. We use the
corresponding curvature matrices to characterize statistical properties of the
recovered parameters. We incorporate in the formalism some of the essential
features of the CMB data sets, such as inhomogeneous pixel domain noise,
unknown map offsets as well as calibration errors and study their consequences
for the separation. We find that the calibration is likely to have a dominant
effect on the precision of the spectral parameter determination for a realistic
CMB experiment. We apply the algorithm to simulated data and discuss the
results. Our focus is on partial-sky, total-intensity and polarization, CMB
experiments such as planned balloon-borne and ground-based efforts, however,
the techniques presented here should be also applicable to the full-sky data as
for instance, those produced by WMAP and anticipated from Planck.
