* Title : Generalized Fréchet Means with Random Domains and Random Cost Functions

* Abstract :

Fréchet means provide a fundamental tool for describing random objects in non-Euclidean spaces. However, many statistical procedures do not fit the classical framework because the parameter space differs from the data space, the empirical optimization domain is data-dependent, or the cost function involves estimated auxiliary parameters. We develop a unified framework of generalized Fréchet means that accommodates these features. We first allow the empirical and population minimizing domains to differ, with the empirical domain possibly random. Under conditions involving Kuratowski convergence of the domains, regularity of the cost function, and eventual compactness of the empirical minimizers, we establish strong consistency of the empirical generalized Fréchet mean set. This result provides a direct theoretical foundation for sequential dimension reduction on non-Euclidean spaces, as illustrated by the strong consistency of principal geodesic analysis on hyperspheres. Related applications include compositional subspace analysis and clustering on metric spaces. We then extend the framework to random cost functions induced by preliminary estimators. This extension is motivated by a complex drift model for ENDOR spectroscopy, in which batch-specific phase and amplitude drifts and an unknown noise covariance create a Neyman–Scott-type inconsistency for joint profile likelihood estimation. A two-step plug-in procedure, combined with the generalized Fréchet mean framework, yields a strongly consistent estimator of the spectral parameter in complex projective space. These results provide a common consistency theory for non-Euclidean estimators involving data-dependent constraints and losses.