  We study the properties of several likelihood-based statistics commonly used
in testing for the presence of a known signal under a mixture model with known
background, but unknown signal fraction. Under the null hypothesis of no
signal, all statistics follow a standard normal distribution in large samples,
but substantial deviations can occur at low sample sizes. Approximations for
respective $p$-values are derived to various orders of accuracy using the
methodology of Edgeworth expansions. Adherence to normality is studied, and the
magnitude of deviations is quantified according to resulting inflation or
deflation. We find that approximations to third-order accuracy are generally
sufficient to guarantee $p$-values with nominal false positive error rates in
the five sigma range ($p$-value $= 2.87 \times 10^{-7}$) for the classic Wald,
score, and likelihood ratio (LR) statistics at relatively low samples. Not only
does LR have better adherence to normality, but it also consistently
outperforms all other statistics in terms of false negative error rates. The
reasons for this are shown to be connected with high-order cumulant behavior
gleaned from fourth order Edgeworth expansions. Finally, a conservative
procedure is suggested for making finite sample adjustments while accounting
for the look elsewhere effect with the theory of random fields (a.k.a. the
Gross-Vitells method).
