Finite Sample Smeariness on Spheres and Modulation-aware Tests

Abstract

Directional and shape data often live on manifolds, where standard central limit theorems (CLT) and the associated Wald-type \(\chi^2\)-tests require curvature-dependent recalibration: the limiting normal covariance is modified by the Hessian of the Fr\'echet function. On spheres, under the positive-Hessian assumptions considered here, this curvature correction yields Type~I finite sample smeariness (FSS), meaning that the asymptotic modulation exceeds \(1\). This paper studies FSS on \(\mathbb{S}^m\) and develops modulation-aware tests. We first show that, for absolutely continuous distributions on \(\mathbb{S}^m\) with positive definite Hessians, Type~I FSS is unavoidable under the stated classical CLT and moment assumptions. The geometric mechanism is that, for the absolutely continuous spherical distributions considered here, positive curvature of the sphere gives the strict spectral bound \(H \prec 2I\), so that the limiting modulation is strictly larger than one. We then prove a curse-of-dimensionality result for dimensionally comparable rotationally symmetric families: their asymptotic modulation is monotone increasing with dimension and can become arbitrarily large as the support approaches a hemisphere. Motivated by these results, we derive \textit{explicit} consistent plug-in estimators of the Fr\'echet-function Hessian and use them to construct one-sample and two-sample Hessian-corrected Hotelling-type statistics with \(\chi^2\) limits. Numerical experiments on low- and high-dimensional spheres show that the proposed modulation-aware tests have rejection behavior comparable to bootstrap-based FSS corrections, while being substantially faster due to the explicit Hessian formulas.