Smoothed least absolute deviation estimation in functional linear model

The functional linear model extends classical regression by modeling scalar responses as functions of stochastic processes. This paper proposes a novel convolution-type smoothed least absolute deviation estimator that addresses the non-smoothness and strict convexity challenges of conventional least absolute deviation estimation. By approximating both the predictor variable and slope function via functional principal component basis expansions, we develop a robust estimator with strong theoretical guarantees. Under mild regularity conditions, we establish the estimator's consistency aligning with the least absolute deviation estimator as the bandwidth vanishes and derive the convergence rate for the prediction error. Simulation studies demonstrate that the proposed smoothed least absolute deviation estimator outperforms conventional estimation methods--including ordinary least squares, standard least absolute deviation, spline-based regression, penalized spline smoothing, and Bayesian estimation, particularly in scenarios involving heavy-tailed error distributions, outlier contamination, and heteroscedasticity. Applications to the Berkeley Growth Study and the Capital Bike Share dataset further validate its practical utility.