On testing the significance of a mode

We propose a nonparametric test for the significance of a mode, with the aim of evaluating whether a region of relatively high observed density reflects the actual presence of a mode in the true distribution underlying a set of data. The method leverages on Morse theory to characterize the local properties of the modes and the gradient. This allows the definition of an asymptotic test, based on the concept of gradient ascent paths and relying on resampling methods, to approximate the distribution of the test statistic under the null hypothesis. The performances of the proposed test statistic and the control of the Type-I error are shown via multiple simulation studies.