Scalar-on-function regression with measurement error in the functional regressors

Abstract: We consider the problem of scalar-on-function regression. Most existing methods implicitly assume that the functional covariates are observed exactly, but in practice, they are often contaminated by measurement error. Our goal, therefore, is to deal with the problem of scalar-on-function regression when the regressor functions are observed with error. In this paper, we propose a simulation-extrapolation method to correct for the attenuation of estimated coefficient functions caused by the error. The method first estimates the error variance, establishes the relationship between a sequence of added error variance and the corresponding estimates of coefficient functions, and then extrapolates to the zero-error. We describe three methods to extrapolate the sequence of estimated coefficient functions. In a simulation study, we compare the performance of the simulation-extrapolation method with two pre-smoothing methods based on smoothing splines and functional principal component analysis. Next, we discuss the extension of the method in several directions, allowing for more complex noise covariance structures, multiple replications of functional predictors, generalized responses, and 2D and 3D functional predictors. Finally, we illustrate the methods by an application to diffusion tensor imaging data.