Bivariate and multivariate distributions with bimodal marginals

On Generalizing Sarle’s Bimodality Coefficient as a Path towards a Newly Composite Bimodality Coefficient

Determining whether a distribution is bimodal is of great interest for many applications. Several tests have been developed, but the only ones that can be run extremely fast, in constant time on any variable-size signal window, are based on Sarle’s bimodality coefficient. We propose in this paper a generalization of this coefficient, to prove its validity, and show how each coefficient can be computed in a fast manner, in constant time, for random regions pertaining to a large dataset. We present some of the caveats of these coefficients and potential ways to circumvent them. We also propose a composite bimodality coefficient obtained as a product of the weighted generalized coefficients. We determine the potential best set of weights to associate with our composite coefficient when using up to three generalized coefficients. Finally, we prove that the composite coefficient outperforms any individual generalized coefficient.