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February 2021 Wasserstein $F$-tests and confidence bands for the Fréchet regression of density response curves
Alexander Petersen, Xi Liu, Afshin A. Divani
Ann. Statist. 49(1): 590-611 (February 2021). DOI: 10.1214/20-AOS1971


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 Fréchet means under a suitable metric. Using the Wasserstein geometry of optimal transport, we consider the Fréchet 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.


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Alexander Petersen. Xi Liu. Afshin A. Divani. "Wasserstein $F$-tests and confidence bands for the Fréchet regression of density response curves." Ann. Statist. 49 (1) 590 - 611, February 2021.


Received: 1 October 2019; Revised: 1 April 2020; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1971

Primary: 62F03, 62F25, 62J99
Secondary: 62F05, 62F12

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • February 2021
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