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.
Categories: math.ST stat.TH