On Functional Data Analysis with Dependent Errors

Abstract

Classical functional data analysis (FDA) is based on directly observed random curves. However, in a more realistic setting such as for certain types of EEG data, the observations are perturbed by noise even strongly dependent noise. In this dissertation the influence of long memory noise on trend and covariance estimation, functional principal component analysis and two sample inference is investigated. Firstly, the kernel estimation of trend function and covariance function in repeated time series with long memory errors is considered. Functional central limit theorem of estimated trend and estimated covariance is established. Since the main quantity of interest in FDA is the covariance, the trend plays the role of a nuisance parameter. Therefore, the orthogonal contrast transformation is proposed to eliminate the trend before estimating the covariance. In order to relax the constrain between the number of random curves and the number of sampling points on each curve, higher order kernels are used. Secondly, we consider the estimation of eigenvalues, eigenfunctions (functional principal components) and functional principal component scores in FDA models with short or long memory errors. It turns out that there is no difference between short and long memory errors when considering the asymptotic distribution of estimated eigenvalues and estimated eigenfunctions. However, the asymptotic distribution of estimated scores and the rate of convergence differ significantly between weakly and strongly dependent errors. Moreover, long memory property not only lead to a slower rate of convergence, but the dependence of score estimators. Thirdly, two sample inference for eigenspaces in FDA models with dependent errors is discussed. A test for testing the equality of subspaces spanned by a finite number of eigenfunctions is constructed and its asymptotic distribution under the null hypothesis is derived. This provides the basis for defining suitable test procedures. In order to obtain asymptotically exact rejection regions, the joint asymptotic distribution of the residual process is required. However, since the dimension of the subspace is in most cases very small, we propose to use a simple Bonferroni adjusted test. A more practical solution is a bootstrap test which is also applicable even for small samples.