Simultaneous Confidence Bands for Functional Regression Models

Abstract

In recent years, the field of functional data analysis (FDA) has received a great deal of attention, and many useful theories and interesting applications have been reported. One topic of particular interest involves estimation of simultaneous confidence bands (SCB) for an unknown function. Degras (2011) proposed an estimator of SCBs for the mean function in a simple (no covariates) function-on-scalar regression model that relies on some assumptions on the tail behavior of the errors. In the case that such distributional assumptions do not hold, Degras also proposed a bootstrap method (sampling with replacement) but did not study its asymptotic properties. We consider a more general function-on-scalar regression model, involving multiple covariates, and allowing the variance function of the functional responses to be dependent on the covariates. In addition, we propose a wild bootstrap method for estimating SCBs for the coefficient function in the more general function-on-scalar regression model with multiple covariates and non-normal errors. We provide some asymptotic theory to demonstrate validity of this procedure for the simple case (no covariates), and for more general models we present results from a simulation study to study finite sample properties of this method and to compare it with the sampling-with-replacement bootstrap method.