Abstract:
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Faced with new types of data, statisticians require models that can handle samples of complex random objects. This motivates the development of regression models that feature random objects in a metric space as responses and scalars or vectors as predictors. For such regression scenarios, we develop generalized versions of both global least squares regression and local weighted least squares smoothing by extending the concept of Frechet means. We derive asymptotic rates of convergence for the corresponding fitted regressions to the population targets and obtain limit distributions for the special case of random objects that reside in a Hilbert space, as encountered for example in functional response models. The resulting regression models have broad applicability and are demonstrated for examples from brain imaging and demography that feature responses ranging from functional data to probability distributions and correlation matrices.
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