Power spectra, estimated by the maximum entropy method and by a fast Fourier transform based periodogram method, are compared using simulated time series. The time series are computer generated by passing Gaussian white noise through low-pass filters with precisely defined magnitude response curves such that the output time series have power law spectra in a limited frequency range: P( f)=Af-p, f1≤ f≤ f2. Ten different values of p between 0.5 and 5.0 are used. Using 4000 independent realizations of these simulated time series, it is shown that maximum entropy results are superior (usually greatly superior) to the periodogram results even when end-matching or windowing or both are used before the power spectra are estimated. Without the use of end-matching or windowing or both, the periodogram results are useless at best and very misleading at worst. A 5-min section of ionospheric scintillation data from the MARISAT satellite has been broken into 60 overlapping sections, each 10 s long and overlapped by 5 s. This section was chosen because it illustrated a transition from low-level background noise to moderate scintillation and another transition to fully saturated scintillation. Order 5 Burg-MEM spectra from the raw data are compared with periodograms computed from end-matched and windowed data. The superiority of Burg-MEM rests largely in the smoothness of the spectrum: real changes in spectral shape are not obscured by meaningless detail.