The Bootstrap for the FunctionalAutoregressive Model FAR(1)

Abstract

Functional data analysis is a branch of statistics that deals with observationsX1,...,Xnwhich are curves. We are interested in particular in time series of dependent curves and,specifically, consider the functional autoregressive process of order one (FAR(1)), whichis defined asXn+1= Ψ(Xn) + n+1with independent innovations t. EstimatesˆΨ for the autoregressive operator Ψ have been investigated a lot during the last two decades, andtheir asymptotic properties are well understood. Particularly difficult and different fromscalar- or vector-valued autoregressions are the weak convergence properties which alsoform the basis of the bootstrap theory.Although the asymptotics forˆΨ(Xn) are still tractable, they are only useful for larg eenough samples. In applications, however, frequently only small samples of data areavailable such that an alternative method for approximating the distribution ofˆΨ(Xn)is welcome. As a motivation, we discuss a real-data example where we investigate achangepoint detection problem for a stimulus response dataset obtained form the animalphysiology group at the Technical University of Kaiserslautern.To get an alternative for asymptotic approximations, we employ the naive or residual-based bootstrap procedure. In this thesis, we prove theoretically and show via simulationsthat the bootstrap provides asymptotically valid and practically useful approximations ofthe distributions of certain functions of the data. Such results may be used to calculateapproximate confidence bands or critical bounds for tests.