The effects of noise in autoregressive (AR) analysis (or linear prediction) and its compensation (NCAR) has been commonly carried out in the time domain under the least square (LS) criterion. This paper studies the adequacy of such an approach by means of a comparative analysis with selected frequency-based NCAR methods. In particular, the maximization of the spectral likelihood (ML) results in a proper optimization problem that is easy to solve and brings useful insights into the rationale of the NCAR problem. On the contrary, popular time-based NCAR methods are shown in the paper to be designed, in the ML context, around ill-conditioned criteria, requiring constraints to guarantee stable solutions. The statistical analysis on a realistic scenario as well as an experiment on speech enhancement complement this analysis.