Inference for Fréchet Mean Sets

Abstract

We construct confidence sets for the entire Fréchet mean set by inverting studentized pairwise risk inequalities. Gaussian multiplier calibration localized around empirical near minimizers gives asymptotic coverage at least the nominal level, uniformly over distribution classes satisfying empirical-process conditions. The construction requires neither unique ness nor a manifold structure. We characterize the limiting binding directions and give conditions for exact asymptotic coverage. On the intrinsic circle, exactness holds for every fixed non-point-mass law and uniformly over a family in which two means merge at a quar tic minimum. With ideal multiplier quantiles, the construction reproduces the Euclidean Wald set under both nonsingular and singular covariance. Under regular local conditions, it is first-order equivalent to intrinsic Wald inference.