Exponential Lp-quantile estimation of function-on-scalar model with its applications in functional data analysis

The function-on-scalar model (FOSM) is a well-known tool used to investigate the relationship between a functional response variable and a set of scalar covariates. In the literature, a substantial body of research on FOSM primarily focuses on classical least squares-based inference, which can yield biased or unreliable results in the presence of outliers or data contamination. In this paper, we propose a robust estimation procedure for FOSM utilizing the exponential $ L^p $ Lp-quantile loss ( $ L^p $ Lp-EQL) with a tuning parameter, and the optimal value for this tuning parameter can be automatically selected based on the observed data. Furthermore, we provide an iterative algorithm to compute the estimates. Under defined regularity conditions, we demonstrate that the proposed estimators exhibit desirable large-sample properties, including estimation consistency and asymptotic normality. Finally, we conduct numerical simulations to evaluate the finite sample performance of the proposed estimators, and we present two real data applications to illustrate the utility of the proposed methods.