### 4.9 Upper limits

Detection of a signal is signified by a large value of the -statistic that is unlikely to arise from the
noise-only distribution. If instead the value of 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 , where is the detection probability given by Equation (49), is the
value of the -statistic corresponding to the loudest event, and is a chosen confidence [15, 1]. Then
is the desired upper limit with confidence .
When gravitational-wave data do not conform to a Gaussian probability density assumed in
Equation (49), a more accurate upper limit can be obtained by injecting the signals into the detector’s data
and thereby estimating the probability of detection [3].