Robust shrinkage estimation and selection for functional multiple linear model through LAD loss

In functional data analysis (FDA), variable selection in regression model is an important issue when there are multiple functional predictors. Most of the existing methods are based on least square loss and consequently sensitive to outliers in error. Robust variable selection procedure is desirable. When functional predictors are considered, both non-data-driven basis (e.g. B-spline) and data-driven basis (e.g. functional principal component (FPC)) are commonly used. The data-driven basis is flexible and adaptive, but it raise some difficulties, since the basis must be estimated from data.Since least absolute deviation (LAD) loss has been proven robust to the outliers in error, we propose in this paper a robust variable selection with data-driven basis FPC and LAD loss function. The asymptotic results are established for both fixed and diverging pp. Our results include the existing results as special cases. Simulation results and a real data example confirm the effectiveness of the proposed method.