A recent letter reports an upper limit on the galactic compact binary inspiral rate set by gravitational wave observations using the LIGO 40M prototype interferometer. The limit was set by imagining a threshold set equal to the amplitude of the loudest event and then considering the probability that only one event was observed with amplitude greater than the threshold. That limit can be improved on by considering also that the observed event took on a particular value. In this communication we derive a general and exact expression for the likelihood that the loudest event, observed with imperfect efficiency and in the presence of a background, has a given amplitude. We apply our result to find an improved upper limit, based on the same assumptions and using only those results described in . Typeset using REVTEX ∗Also Department of Physics and Department of Astronomy and Astrophysics; e-mail address LSF5@PSU.Edu.