Single Index Fréchet Regression for Random Object Responses

• Activity Number: 267 - Nonparametric Statistics Student Paper Competition Presentations
• Type: Topic Contributed
• Date/Time: Tuesday, August 9, 2022 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Satarupa Bhattacharjee* and Hans-Georg Müller
• Companies: University of California, Davis and University of California, Davis
• Keywords: Single-index model; Dimension reduction; Random objects; M-estimation; Non-Euclidean data; Inference

Abstract

The increasing abundance of complex non-Euclidean data not lying in a vector space motivates the development of suitable statistical methods. An essential aspect is to build regression methodologies to quantify the dependence of a general metric space valued response variable on Euclidean predictors, by modeling the conditional Fréchet means. The responses include covariance matrices, graph Laplacians of networks, and univariate probability distribution functions among other complex objects that lie in abstract metric spaces. A novel single-index model is proposed for metric space-valued random object responses coupled with multivariate Euclidean predictors to facilitate semiparametric inference for the index parameter. We show here that for the case of multivariate Euclidean predictors, the parameters that define a single index projection vector can be used to substitute for the inherent absence of parameters in Fréchet regression. We derive the asymptotic distribution of suitable estimates of these parameters and utilize them to test linear hypotheses for the parameters, subject to an identifiability condition. The method is illustrated for resting-state FMRI data.