Nonparametric statistical inference under complex temporal dynamics

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The last two decades have witnessed an enormous increase in the need for analyzing data with complex temporal dynamics. In particular, two types of time series data structures are of drastically growing interests in both theory and practice. First, it is evident that the temporal dependence structures and marginal distributions of many time series data change both abruptly and smoothly over time. Second, for many long and densely observed time series data, it is mathematically elegant and beneficial to separate the time record into natural consecutive intervals and treat the data as functional time series. ***The proposed research is aimed at providing a systematic package of statistical methodologies and theory for the modelling, estimation, inference and prediction of the aforementioned two types of complex structured time series in a unified framework of nonlinear system representation. Methodologically, we will develop efficient and adaptive nonparametric procedures for the analysis of general classes of non-stationary and/or functional valued time series with rigorous mathematical justifications. These include but are not limited to robust, multiscale and adaptive abrupt change detection in non-stationary (functional) time series; efficient and robust simultaneous confidence bands for nonparametric regression of non-stationary time series; uniform inference of both sparsely and densely observed functional time series and efficient  simultaneous nonparametric inference of time-varying spectral densities. Theoretically, a systematic statistical theory for non-stationary time series with diverging dimensionality will be developed, which provides a mathematical foundation for most of the research topics mentioned above. In particular, systematic empirical process theory and Gaussian approximation theory will be established for a wide class of non-stationary time series with diverging dimensionalities in the proposed research. ***Nowadays, technological innovations have made it possible to collect massive amount of data with complex structures over a relatively long period of time. I see a great demand, opportunity and challenge for statistical analysis of non-stationary time series with or without functional forms. Indeed, statistical theory and methodologies should progress with the trend of the data. However, a unified statistical theory for non-stationary time series analysis is still lacking due to the lack of appropriate statistical and probabilistic tools. I believe that the proposed framework from the nonlinear system point of view will provide an important theoretical and methodological basis for non-stationary (functional) time series analysis in many scientific disciplines.**