Frechet single index models for object response regression

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 R-p and the response variables are situated in a metric space, conditional Frechet means can be used to define the Frechet regression function. Global and local Frechet 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 Frechet single index model, in which the Frechet regression function is assumed to depend only on a scalar projection of the multivariate predictor. Estimation is performed by combining local Frechet 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.