|
Abstract:
|
We introduce a single-index model for regression models where metric space-valued random object responses are coupled with multivariate Euclidean predictors. The responses include complex, non-Euclidean data such as covariance matrices, graph Laplacians of networks or univariate probability distribution functions. While Fréchet regression has proved useful for modeling the conditional Fréchet mean of such random objects given multivariate Euclidean vectors, it does not provide for regression parameters such as slopes or intercepts, since the metric space-valued responses are not amenable to linear operations. As a consequence, distributional results for Fréchet regression have been elusive. We show that the parameters that define a single index and projection vector can be used to substitute for the inherent absence of parameters in Fréchet regression and then proceed to demonstrate asymptotic convergence results and inference for suitable estimates of these parameters as well as consistent estimation of the link function. The method is illustrated with simulations and an application to resting state fMRI data.
|
