Wasserstein $F$-tests and Confidence Bands for the Fr\`echet Regression of Density Response Curves

TL;DR

This paper develops Wasserstein-based statistical tests and confidence bands for Fréchet regression of density curves, enabling inference on the relationships between densities and predictors in a nonlinear space.

Contribution

It introduces novel Wasserstein geometry tools for Fréchet regression, including tests and confidence bands, with theoretical guarantees and practical validation.

Findings

Tests have correct size and power in simulations.

Confidence bands achieve nominal coverage.

Method applied successfully to clinical density data.

Abstract

Data consisting of samples of probability density functions are increasingly prevalent, necessitating the development of methodologies for their analysis that respect the inherent nonlinearities associated with densities. In many applications, density curves appear as functional response objects in a regression model with vector predictors. For such models, inference is key to understand the importance of density-predictor relationships, and the uncertainty associated with the estimated conditional mean densities, defined as conditional Fr\'echet means under a suitable metric. Using the Wasserstein geometry of optimal transport, we consider the Fr\'echet regression of density curve responses and develop tests for global and partial effects, as well as simultaneous confidence bands for estimated conditional mean densities. The asymptotic behavior of these objects is based on underlying functional central limit theorems within Wasserstein space, and we demonstrate that they are asymptotically of the correct size and coverage, with uniformly strong consistency of the proposed tests under sequences of contiguous alternatives. The accuracy of these methods, including nominal size, power, and coverage, is assessed through simulations, and their utility is illustrated through a regression analysis of post-intracerebral hemorrhage hematoma densities and their associations with a set of clinical and radiological covariates.