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Abstract:
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Existing work on functional response regression has focused on mean regression. In this paper, we study function-on-scalar quantile regression (FQR), which can provide a comprehensive understanding of how scalar predictors influence the entire distribution of functional responses. We introduce a scalable, distributed strategy to perform FQR that can account for intrafunctional correlations in functional responses. This general distributed strategy first performs separate quantile regression to compute M-estimators at each sampling location, and then carries out estimation and inference for the entire coefficient functions by properly exploiting the uncertainty quantifications and dependence structures of M-estimators. We derive a uniform Bahadur representation and a strong Gaussian approximation result for the M-estimators on the discrete sampling grid, which are of independent interest and provide theoretical justification for this distributed strategy. Some large sample properties of our proposed coefficient function estimators are described. We conduct simulations to assess the finite sample performance of the proposed methods and apply to a mass spectrometry proteomics dataset.
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