Maximum Likelihood Fitting of X-Ray Power Density Spectra: Application to High-frequency Quasi-periodic Oscillations from the Neutron Star X-Ray Binary 4U1608-522

High-frequency quasi-periodic oscillations (QPOs) from weakly magnetized neutron stars display rapid frequency variability (second timescales) and high coherence with quality factors up to at least 200 at frequencies about 800-850 Hz. Their parameters have been estimated so far from standard min(χ²) fitting techniques, after combining a large number of power density spectra (PDS), to have the powers normally distributed (the so-called Gaussian regime). Before combining PDS, different methods to minimize the effects of the frequency drift to the estimates of the QPO parameters have been proposed, but none of them relied on fitting the individual PDS. Accounting for the statistical properties of PDS, we apply a maximum likelihood method to derive the QPO parameters in the non-Gaussian regime. The method presented is general, easy to implement, and can be applied to fitting individual PDS, several PDS simultaneously, or their average, and is obviously not specific to the analysis of kHz QPO data. It applies to the analysis of any PDS optimized in frequency resolution and for low-frequency variability or PDS containing features whose parameters vary on short timescales, as is the case for kHz QPOs. It is equivalent to the standard χ² minimization fitting when the number of PDS fitted is large. The accuracy, reliability, and superiority of the method is demonstrated with simulations of synthetic PDS, containing Lorentzian QPOs of known parameters. Accounting for the broadening of the QPO profile, due to the leakage of power inherent to windowed Fourier transforms, the maximum likelihood estimates of the QPO parameters are asymptotically unbiased and have negligible bias when the QPO is reasonably well detected. By contrast, we show that the standard min(χ²) fitting method gives biased parameters with larger uncertainties. The maximum likelihood fitting method is applied to a subset of archival Rossi X-ray Timing Explorer data of the neutron star X-ray binary 4U1608-522, for which we show that the lower kHz QPO parameters can be measured on timescales as short as 8 s. To demonstrate the potential use of the results of the maximum likelihood method, we show that in the observation analyzed the time evolution of the frequency is consistent with a random walk. We then show that the broadening of the QPO due to the frequency drift scales as \sqrt{T}, as expected from a random walk (T is the integration time of the PDS). This enables us to estimate the intrinsic quality factor of the QPO to be ~260, whereas previous analysis indicated a maximum value around 200.