Proposal: DMS 9504798 PI(s): Kung-Sik Chan and Osnat Stramer Institution: University of Iowa Title: Nonlinear Modeling in Continous Time, Delayed Autoregressive Processes, and Chaos Abstract: In analyzing nonlinear time series data sampled irregularly, it is natural to consider non-linear continuous-time models for the (unobserved) underlying continuous-time process. Owing to the intractability of the likelihood function for such kind of nonlinear models, the existing estimation methods in the literature are somewhat ad hoc. This research contributes to the statistical inference for non-linear continuous-time modeling based on (possibly irregularly sampled) discrete-time observations. This research consists of four parts. In part I, the investigators study maximum likelihood estimation for the general class of nonlinear continuous-time autoregressive process, abbreviated by NLCAR(p). The investigators derive a new expression for the likelihood function which can be evaluated via simulation. The investigators implement the simulation-based maximum likelihood estimation procedure for some classes of NLCAR(p) models. The investigators then study the sampling properties of the maximum likelihood estimator so obtained. In part II, the investigators apply the Lagrange Multiplier tests for detecting non-linearity and also those for other diagnostic purposes, and study their large sample properties. In part III, the investigators extend an approach relating ergodicity and stability to continuous-time models. In part IV, the investigators study delayed continuous-time non-linear autoregressive processes whose state space is an infinite-dimensional Hilbert space. They also study the recurrence (stationarity) properties and the statistical inference for the delayed non-linear autoregressive processes. In this research, the investigators study new statistical methods useful for analyzing data which are taken over possibly unequal ti me intervals. Such kind of data is known as irregularly sampled time series, and occurs frequently in diverse areas including business forecasting, environmental statistics, medical statistics, physical and engineering science. The investigators study computer-intensive methods which provide the relevant tools for studying and testing for the nonlinear structure of irregularly sampled time series data. The investigators also develop some mathematical statistics and probability theory fundamental to the understanding of the new methods. LEVEL OF EFFORT STATEMENT At the recommended level of support, the PI will make every attempt to meet the original scope and level of effort of the project.
