The proposed research provides novel methodology for the statistical inference of functional and high-dimensional time series with a focus on multi-sample functional data and panel data. The investigators thereby extend the range of functional data analysis beyond the currently available models. This is relevant because the proposed models have a number of important applications in economics and finance, geophysics and engineering. A characteristic common to many functional data applications is the presence of dependence. The theory of functional data, however, is as of today mainly focused on independent processes with notable exceptions being given by the functional autoregressive and linear processes. This proposal represents a comprehensive research plan for developing an extended tool-kit for the analysis of functional and high-dimensional time series data. It contains the following parts:development of fully functional tests for independence and stationarity, and diagnostic tests that do not require dimension reduction; advancement of functional principal component analysis by taking into account that different operators should be used under the null and alternative hypothesis and by providing novel theory for the case of an increasing number of principal components; introduction and development of the theoretical foundation of the concept of functional analysis of variance, including procedures to cluster functional time series observations into groups; advancement of the methodology of panel data to more general models, explicitly allowing for the high-dimensionality of the observations but notably not requiring stationarity of the panels; and the breaking of new ground by combining functional series with high-dimensional time series methodology. This requires the development of sophisticated new statistical methodology, including the refinement and extension of the theory of (vector-valued) Hilbert space-valued observations. The research also includes a significant innovative computational component. To aid the dissemination of results, we plan to make the relevant software freely available via the Internet. Completion of the proposal gives statisticians and practitioners new tools for analyzing different forms of functional data.
The proposal is interdisciplinary in nature, with applications in diverse fields ranging from finance and economics (tick-by-tick transaction data, joint movement of several economic indicators, the effect of policy changes on economic processes), environmental science (monitoring air pollution, changes in temperature, change in the occurrences of certain meteorological extremes), and to geophysics (magnetic field readings of magnetometers). In the context of financial data, independence and stationarity testing can be used to determine if, for example, the functional autoregressive model is appropriate for high resolution asset price data. If so then further estimation techniques can be applied towards predicting asset values as well as other techniques in economic forecasting. By applying the functional analysis of variance to magnetic field measurements taken from several different locations one may categorize these locations according to the magnetic field behavior they exhibit. This may influence the implementation of radio communication in these areas. The research is therefore of immediate interest for practitioners and will further connect statistics and fields of science with a significant statistical component. It also advances the theory of mathematical statistics. The proposed research produces doctoral students, among them female and minority students, theoretically and practically versed in both statistics and an area of application. The training and involvement of undergraduate students in this research is also included through regular coursework, independent study and projects.
