SpectrumSolvers™ (free) program helps Find best Spectral Estimation Method for a Power Spectral Density plot. A menu of 10+ spectral estimators from Steven Kay's textbook 'Modern Spectral Estimation' 1988 is available to choose from. The results differ dramatically from one estimator to another. Plus, varying input parameters and/or number of points may provide discrepancies. Observe how zero padding effects your results. Manufacturing companies take note! Some estimators can detect signals 50 to 100 dB from main signal. See documented example! The unwritten rule of '30 dB is okay' (i.e. hidden) is no longer true.
Visit https://www.quora.com/Spectral-Estimation-Algorithms-Methods-whats-available for more on topic.
At http://fortrancalculus.info/apps/spectrumsolvers.html the (Windows) App may be download it, install it, and run some demos
The multitaper method sort of optimizes preservation of the time series and the approximation to the spectrum (good bias-variance trade off, see the attached .pdf and references therein )