Local Null Hypothesis Significance Testing on Riemaniann Manifolds

We discuss the problem of testing hypotheses on functional data defined on manifold domain. Aim of this work is to test hypotheses locally, i.e., to provide a p-value function (p) over tilde (s) that can be used to test a null hypothesis against an alternative one on every point of the manifold. In the following we present three methods to do so: an adjusted e-value function controlling the functional false discovery rate; an adjusted p-value function controlling the ball-wise error rate; an adjusted e-value function that is ball-wise e-valid.