Robust estimation for a general functional single index model via quantile regression

This paper studies the estimation of a general functional single index model, in which the conditional distribution of the response depends on the functional predictor via a functional single index structure. We find that the slope function can be estimated consistently by the estimation obtained by fitting a misspecified functional linear quantile regression model under some mild conditions. We first obtain a consistent estimator of the slope function using functional linear quantile regression based on functional principal component analysis, and then employ a local linear regression technique to estimate the conditional quantile function and establish the asymptotic normality of the resulting estimator for it. The finite sample performance of the proposed estimation method is studied in Monte Carlo simulations, and is illustrated by an application to a real dataset.