Nonparametric Local Tests for Functional Data on Manifold Domains

We introduce a local inferential framework for functional data defined on a Riemannian manifold, extending existing methods for one-dimensional domains. Our approach focuses on pointwise hypothesis testing while addressing multiple testing concerns through two distinct error control strategies: false discovery rate (fFDR) control and ball-wise error rate (BWER) control. We demonstrate the effectiveness of these methods using a numerical example on functional data defined on a spherical domain. The results highlight the strengths of each approach in detecting significant regions while maintaining rigorous error control.