  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.
