Functional Group Bridge for Simultaneous Regression and Support Estimation
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There is a wide variety of applications where the unknown nonparametric functions are locally sparse, and the support of a function as well as the function itself is of primary interest. In the function-on-scalar setting, while there has been a rich literature on function estimation, the study of recovering the function support is limited. In this article, we consider the function-on-scalar mixed effect model and propose a weighted group bridge approach for simultaneous function estimation and support recovery, while accounting for heterogeneity present in functional data. We use B-splines to transform sparsity of functions to its sparse vector counterpart of increasing dimension, and propose a fast non-convex optimization algorithm for estimation. We show that the estimated coefficient functions are rate optimal in the minimax sense under the L_2 norm and resemble a phase transition phenomenon. For support estimation, we derive a convergence rate under the L_∞ norm that leads to a sparsistency property under δ-sparsity, and provide a simple sufficient regularity condition under which a strict sparsistency property is established. An adjusted extended Bayesian information criterion is proposed for parameter tuning. The developed method is illustrated through simulation and an application to a novel intracranial electroencephalography (iEEG) dataset.
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