  A {\it pure significance test} (PST) tests a simple null hypothesis
$H_f:Y\sim f$ {\it without specifying an alternative hypothesis} by rejecting
$H_f$ for {\it small} values of $f(Y)$. When the sample space supports a proper
uniform pmf $f_\mathrm{unif}$, the PST can be viewed as a classical likelihood
ratio test for testing $H_f$ against this uniform alternative. Under this
interpretation, standard test features such as power, Kullback-Leibler
divergence, and expected $p$-value can be considered. This report focuses on
PSTs for multinomial and binomial distributions, and for the related
goodness-of-fit testing problems with the uniform alternative. The case of
repeated observations cannot be reduced to the single observation case via
sufficiency. The {\it ordered binomial distribution}, apparently new, arises in
the course of this study.
