The Statistics of the Cross-Spectrum and the Spectrum Average: Generalization to Multiple Instruments

This article addresses the measurement of the power spectrum of red noise processes at the lowest frequencies, where the minimum acquisition time is so long that it is impossible to average on a sequence of data record. Therefore, averaging is possible only on simultaneous observation of multiple instruments. This is the case of radio astronomy, which we take as the paradigm, but examples may be found in other fields such as climatology and geodesy. We compare the Bayesian confidence interval of the red noise parameter using two estimators, the spectrum average and the cross-spectrum. While the spectrum average is widely used, the cross-spectrum using multiple instruments is rather uncommon. With two instruments, the cross-spectrum estimator leads to the Variance-Gamma distribution. A generalization to q devices based on the Fourier transform of characteristic functions is provided, with the example of the observation of millisecond pulsars with five radio telescopes (RTs). The simulations show that the spectrum average is by a small amount more efficient than the cross-spectrum, chiefly when the background exceeds the signal. However, some notable differences between their upper limit indicate that it should be wiser to compute both estimators.