Concerning parameter estimation using the cosmic microwave background

Most parameter constraints obtained from cosmic microwave background (CMB) anisotropy data are based on power estimates and rely on approximate likeli hood functions; computational difficulties generally preclude an exact anal ysis based on pixel values. With the specific goal of testing this kind of approach, we have performed a complete (un-approximated) likelihood analysi s combining the COBE, Saskatoon and MAX data sets. We examine in detail the ability of certain approximate techniques based on band-power estimates to recover the full likelihood constraints. The traditional chi (2)-method do es not always find the same best-fit model as the likelihood analysis (a bi as), due mainly to the false assumption of Gaussian likelihoods that makes the method overly sensitive to data outliers. Although an improvement, othe r approaches employing non-Gaussian flat-band likelihoods do not always fai thfully reproduce the complete likelihood constraints either; not even when using the exact flat-band likelihood curves. We trace this to the neglect of spectral information by simple flat band-power estimates. A straightforw ard extension incorporating a local effective slope (of the power spectrum, C-l) provides a faithful representation of the likelihood surfaces without significantly increasing computing cost. Finally, we also demonstrate that the best-fit model to this particular data set is a good Ft, or that the o bservations are consistent with Gaussian sky fluctuations, according to our statistics.