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Abstract:
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With the increasing prominence of non-Euclidean data objects, statisticians must develop appropriate statistical tools for their analysis. For regression models with predictors in $R^p$ and response variables being situated in a metric space, conditional Fréchet means can be used to define the Fréchet regression function. Global and local Fréchet methods have recently been developed for modeling and estimating this regression function as extensions of multiple and local linear regression, respectively. In this presentation, this line of methodology is expanded to include the Fréchet Single Index (FSI) model, in which the Fréchet regression function only depends on a scalar projection of the underlying multivariate predictor. Estimation is performed by combining local Fréchet regression along with $M$-estimation to estimate the coefficient vector 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-at-death, viewed as elements of Wasserstein space.
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