Pure Significance Tests for Multinomial and Binomial Distributions: the Uniform Alternative

Abstract: 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.