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Abstract:
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A challenging aspect of the analysis of random objects is the highly-varied and complex geometry of the spaces in which these objects lie. Recently, the concept of Fréchet regression was introduced, generalizing ordinary least squares regression to the case of random object responses. This flexible extension results in a global model that requires no smoothing parameter to fit, thus potentially lending itself to conduct standard inferential procedures in this complex setting. When only a metric is assumed for the response space, the concept of a residual fades away, rendering usual inferential procedures obsolete. Beginning with the global hypothesis of no effect, we develop a test based on a generalized Fréchet $R^2$ coefficient, and derive its limiting null distribution. The performance of the test is then studied through simulation and real data analysis.
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