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

TL;DR

This paper introduces pure significance tests (PSTs) for multinomial and binomial distributions that do not require specifying an alternative hypothesis, focusing on uniform alternatives and exploring their properties and new related distributions.

Contribution

It develops PSTs for multinomial and binomial cases with uniform alternatives, introduces the concept of ordered binomial distribution, and analyzes their statistical properties.

Findings

PSTs can be viewed as likelihood ratio tests against uniform alternatives.

The ordered binomial distribution is a new concept arising in this context.

Standard test features like power and divergence are analyzed for PSTs.

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.