Uniform Convergence of Generalized Conditional Fr\'echet Means with Applications to Weighted Fr\'echet Aggregation and Exceedance Set Estimation

The statistical analysis of object oriented data in non-Euclidean spaces heavily relies on generalized conditional Fréchet means, notably in the context of Fréchet regression.

However, establishing the uniform convergence of these estimators presents several theoretical challenges.

The difficulties are caused primarily by the absence of linear structures in general metric spaces, rendering standard techniques for verifying the asymptotic uniform equicontinuity of the estimator largely intractable.

To overcome this limitation, this paper introduces an alternative theoretical framework for establishing uniform convergence that bypasses the need to verify uniform equicontinuity, under a novel structural condition on the empirical cost function of the generalized conditional Fréchet means.

We demonstrate that this analytical condition is satisfied by various prominent Fréchet regression models across broad classes of metric spaces.

Leveraging these foundational uniform convergence guarantees, we subsequently extend two widely used frameworks from Euclidean to non-Euclidean spaces: (i) a weighted Fréchet aggregation framework that facilitates both distributed regression and robust median-of-means regression; and (ii) an exceedance set estimation framework to identify critical covariate regions where the conditional generalized Fréchet mean surpasses a prescribed threshold, alongside a metric to quantify the aggregate magnitude of the exceedance.

The theoretical properties of these proposed methods are empirically validated through Monte Carlo simulations and an application to dynamic transportation networks in New York City.