Two sides of the same coin: the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -statistic and the 5-vector method

Abstract This work explores the relation between two data-analysis methods used in the search for continuous gravitational waves in LIGO-Virgo-KAGRA data: the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -statistic and the 5-vector method. We show that the 5-vector method can be derived from a maximum likelihood framework similar to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -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.