Robust estimation and variable selection for function‐on‐scalar regression

Abstract

Function-on-scalar regression is commonly used to model the dynamic behaviour of a set of scalar predictors of interest on the functional response. In this article, we develop a robust variable selection procedure for function-on-scalar regression with a large number of scalar predictors based on exponential squared loss combined with the group smoothly clipped absolute deviation regularization method. The proposed procedure simultaneously selects relevant predictors and provides estimates for the functional coefficients, and achieves robustness and efficiency using tuning parameters selected by a data-driven procedure. Under reasonable conditions, we establish the asymptotic properties of the proposed estimators, including estimation consistency and the oracle property. The finite-sample performance of the proposed method is investigated with simulation studies. The proposed method is also demonstrated with a real diffusion tensor imaging data example.

Estimation and Variable Selection under the Function-on-scalar Linear Model with Covariate Measurement Error

“…In the absence of measurement error,Chen et al (2016) andWang et al (2007) established faster convergence rates for using spline methods than (5.24), but they ignored spline approximation error by treating the original function-on-scalar model as a parametric model. On the contrary, the L 2 convergence rate, established byCai et al (2022) for robust function-on-scalar linear regression, is slower than (5.24).The next theorem presents the selection consistency and the point-wise limiting distribution. Define Σ ε = Cov(ε i (t)), where ε i (t) is the error vector given in the model (2.2).…”

“…Parodi and Reimherr (2018) developed a functional linear adaptive mixed estimation procedure to simultaneously obtain sparsity and smoothness for the coefficient functions. Cai et al (2022) considered a robust selection approach with the exponential squared loss function employed.…”