Hypothesis Testing for High-Dimensional Multinomials: A Selective Review

TL;DR

This paper reviews recent methods for hypothesis testing in high-dimensional multinomial distributions, highlighting limitations of traditional tests and proposing minimax-based approaches for more powerful, practical solutions.

Contribution

It introduces a minimax perspective to develop and analyze hypothesis tests that outperform traditional methods in high-dimensional multinomial settings.

Findings

Traditional tests like chi-squared have poor power in high dimensions.

Minimax-based tests can achieve higher power even with non-Normal null distributions.

Refined approaches lead to more practical and effective hypothesis testing methods.

Abstract

The statistical analysis of discrete data has been the subject of extensive statistical research dating back to the work of Pearson. In this survey we review some recently developed methods for testing hypotheses about high-dimensional multinomials. Traditional tests like the $\chi^2$ test and the likelihood ratio test can have poor power in the high-dimensional setting. Much of the research in this area has focused on finding tests with asymptotically Normal limits and developing (stringent) conditions under which tests have Normal limits. We argue that this perspective suffers from a significant deficiency: it can exclude many high-dimensional cases when - despite having non Normal null distributions - carefully designed tests can have high power. Finally, we illustrate that taking a minimax perspective and considering refinements of this perspective can lead naturally to powerful and practical tests.