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

arXiv: https://arxiv.org/abs/1910.13418 · HTML: https://mlsanity.com/paper/1910.13418/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2019.

## Built on by (most cited first, 9 of 9)

- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Hypothesis testing for matched pairs with missing data by maximum mean discrepancy: An application to continuous glucose monitoring](https://mlsanity.com/paper/2206.01590/) (2022)
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- [Energy Trees: Regression and Classification With Structured and Mixed-Type Covariates](https://mlsanity.com/paper/2207.04430/) (2022)
- [Wasserstein Distributional Learning](https://mlsanity.com/paper/2209.04991/) (2022)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [Toward an Asymptotic Efficiency Theory on Regular Parameter Manifolds](https://mlsanity.com/paper/2510.13703/) (2025)
- [Random-Effects Algorithm for Random Objects in Metric Spaces](https://mlsanity.com/paper/2605.02693/) (2026)
- [Deep Single-Index Fr'echet Regression](https://mlsanity.com/paper/2606.06957/) (2026)

## Abstract (arXiv)

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.