Double Folding Test of Unimodality with an Application to Invariant Coordinate Selection

Testing for unimodality is of particular interest in clustering, where multimodality can be viewed as evidence of an underlying group structure. The Double Folding Test of Unimodality (DFTU) was recently proposed as an extension of the Folding Test of Unimodality (FTU), but its original bivariate test statistic does not yield a straightforward global p-value. In this paper, we introduce a new test statistic for the DFTU that admits a well-defined p-value. We derive its asymptotic distribution, leading to a fast approximation of the p-value that avoids computationally intensive Monte Carlo procedures. We then apply the resulting test to component selection in invariant coordinate selection (ICS), a dimension reduction method. The aim is to identify the subspace containing the relevant clustering information and thereby improve the performance of subsequent clustering procedures. In this setting, multimodality offers a natural criterion for identifying informative components, as an alternative to the commonly used non-normality criterion. Applications show that the proposed unimodality-based criterion outperforms the normality-based criterion in identifying the subspace of interest in clustering settings.