  Spherical and hyperspherical data are commonly encountered in diverse applied
research domains, underscoring the vital task of assessing independence within
such data structures. In this context, we investigate the properties of test
statistics relying on distance correlation measures originally introduced for
the energy distance, and generalize the concept to strongly negative definite
kernel-based distances. An important benefit of employing this method lies in
its versatility across diverse forms of directional data, enabling the
examination of independence among vectors of varying types. The applicability
of tests is demonstrated on several real datasets.
