Upper limits

4.8 Upper limits

Detection of a signal is signified by a large value of the $ℱ$ -statistic that is unlikely to arise from the noise-only distribution. If instead the value of $ℱ$ is consistent with pure noise with high probability we can place an upper limit on the strength of the signal. One way of doing this is to take the loudest event obtained in the search and solve the equation

for signal-to-noise ratio $ρUL$ , where $PD$ is the detection probability given by Eq. (87

), $ℱL$ is the value of the $ℱ$ -statistic corresponding to the loudest event, and $β$ is a chosen confidence [23 , 2 ]. Then $ρUL$ is the desired upper limit with confidence $β$ .

When gravitational-wave data do not conform to a Gaussian probability density assumed in Eq. (87 ), a more accurate upper limit can be obtained by injecting the signals into the detector’s data and thereby estimating the probability of detection $PD$ [4 ].