Functional group bridge for simultaneous regression and support estimation

This paper is motivated by studying differential brain activities to multiple experimental condition presentations in intracranial electroencephalography (iEEG) experiments. Contrasting effects of experimental conditions are often zero in most regions and nonzero in some local regions, yielding locally sparse functions. Such studies are essentially a function‐on‐scalar regression problem, with interest being focused not only on estimating nonparametric functions but also on recovering the function supports. We propose a weighted group bridge approach for simultaneous function estimation and support recovery in function‐on‐scalar mixed effect models, 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 nonconvex optimization algorithm using nested alternative direction method of multipliers (ADMM) for estimation. Large sample properties are established. In particular, we show that the estimated coefficient functions are rate optimal in the minimax sense under the L2 norm and resemble a phase transition phenomenon. For support estimation, we derive a convergence rate under the L∞$L_{\infty }$ norm that leads to a selection consistency property under δ‐sparsity, and obtain a result under strict sparsity using a simple sufficient regularity condition. An adjusted extended Bayesian information criterion is proposed for parameter tuning. The developed method is illustrated through simulations and an application to a novel iEEG data set to study multisensory integration.