A Note on the Folding Test of Unimodality: limitation and improved alternative

TL;DR

This paper identifies limitations of the Folding Test of Unimodality in certain mixture models and introduces a Double Folding Test that overcomes these issues, enhancing multimodality detection.

Contribution

The authors analyze FTU failures in mixture models and propose a novel double-folding method that improves unimodality testing accuracy.

Findings

FTU can misclassify multimodal distributions as unimodal in specific settings.

Double Folding Test resolves FTU failures and enhances detection power.

Simulations demonstrate improved performance of the new test.

Abstract

This note addresses a key limitation of the Folding Test of Unimodality (FTU). In specific univariate mixture settings, the folding-based criterion can systematically fail, misclassifying clearly multimodal distributions as unimodal. We fully characterize these failures for Dirac mixtures and extend the analysis to Gaussian mixtures. We then introduce a double-folding procedure that captures complementary information, leading to a new test, the Double Folding Test of Unimodality. It resolves the FTU failures and improves multimodality detection power in simulations.