Is there a statistical hypothesis test that uses the mode? Is there a statistical hypothesis test that considers the mode rather than the mean or median?
 A: After 40+ years, I've never seen a test using the mode in the same way that means (t-tests, anova) or medians (Mann-Whitney) are used to compare between or within groups. I suppose we could conjure up a test of proportions using the modes from two or more groups as a starting point.
A: I concur with Matt Olson in that a hypothesis test comparing two modes between groups will not be akin to existing tests for means or medians (such as ANOVA or ranked tests). This is because with the latter, we can view the statistics as one-dimensional, numeric values, whereas with the mode, we need to look at both its value and frequency. I am not aware of any test comparing modes, as such a procedure would involve the assignment of weightings across bandwidths, and the resulting test statistic would essentially be the solution of an optimisation problem (comparing two KDEs).
Despite this, statistical tests assessing modes in one group have been formulated and put into practical use. These multimodality tests primarily use bootstrapping, or kernel density estimation as a way to smoothen a histogram or a PDF. For instance, Minotte (1997) [1] gave a construction for determining the existence of individual modes of a distribution and asymptotics for their test statistics, which is a notable development from the works of Silverman (1983)[2]. More recently, several of these tests have been implemented in Python and R packages, such as diptest for assessing unimodality.

 References 
 [1] Minotte, M. C. (1997). Nonparametric testing of the existence of modes. Annals of Statistics. 25(4):1646-1660. 
 [2] Silverman, B. W. (1983). Some properties of a test for multimodality based on kernel density estimates. Probability, statistics and analysis. 79:248-259. 
