Fr\'echet single index models for object response regression

Abstract

With the increasing availability of non-Euclidean data objects, statisticians are faced with the task of developing appropriate statistical methods for their analysis. For regression models in which the predictors lie in $\mathbb{R}^p$ and the response variables are situated in a metric space, conditional Fr\'echet means can be used to define the Fr\'echet regression function. Global and local Fr\'echet methods have recently been developed for modeling and estimating this regression function as extensions of multiple and local linear regression, respectively. This paper expands on these methodologies by proposing the Fr\'echet Single Index model, in which the Fr\'echet regression function is assumed to depend only on a scalar projection of the multivariate predictor. Estimation is performed by combining local Fr\'echet along with M-estimation to estimate both the coefficient vector and the underlying regression function, and these estimators are shown to be consistent. The method is illustrated by simulations for response objects on the surface of the unit sphere and through an analysis of human mortality data in which lifetable data are represented by distributions of age-of-death, viewed as elements of the Wasserstein space of distributions.