Two sides of the same coin: the F-statistic and the 5-vector method

This work explores the relationship between two data-analysis methods used in the search for continuous gravitational waves in LIGO-Virgo-KAGRA data: the $\mathcal{F}$-statistic and the 5-vector method. We show that the 5-vector method can be derived from a maximum likelihood framework similar to the $\mathcal{F}$-statistic. Our analysis demonstrates that the two methods are statistically equivalent, providing the same detection probability for a given false alarm rate. We extend this comparison to multiple detectors, highlighting differences from the standard approach that simply combines 5-vectors from each detector. In our maximum likelihood approach, each 5-vector is weighted by the observation time and sensitivity of its respective detector, resulting in efficient estimators and analytical distributions for the detection statistic. Additionally, we present the analytical computation of sensitivity for different searches, expressed in terms of the minimum detectable amplitude.