Doctor of Philosophy

dissertationThis dissertation aims to develop the theory and applications of functional time series analysis. Functional data analysis came into prominence in the 1990s when more sophisticated data collection and storage systems became prevalent, and many of the early developments focused on simple random samples of curves. However, a common source of functional data is when long, continuous records are broken into segments of smaller curves. An example of this is geologic and economic data that are presented as hourly or daily curves. In these instances, successive curves may exhibit dependencies which invalidate statistical procedures that assume a simple random sample. The theory of functional time series analysis has grown tremendously in the last decade to provide methodology for such data, and researchers have focused primarily on adapting methods available in finite dimensional time series analysis to the function space setting. As a first problem, we consider an invariance principle for the partial sum process of stationary random functions. This theory is then applied to the problems of testing for stationarity of a functional time series and the one-way functional analysis of variance problem under dependence

A plug-in bandwidth selection procedure for long run covariance estimation with stationary functional time series

In arenas of application including environmental science, economics, and medicine, it is increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long run covariance function or operator, which is analogous to the long run covariance matrix familiar to finite dimensional time series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies crucially on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite dimensional setting, we propose a bandwidth selection method that aims to minimize the estimator’s asymptotic mean squared normed error (AMSNE) in L2[0, 1]2. As the AMSNE depends on unknown population quantities including the long run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.Natural Sciences and Engineering Research Council of Canad