Asymptotic Theory of Estimation and Testing for Stochastic Processes
In classical time series analysis the asymptotic estimation and testing theory was developed for linear processes, which include the AR, MA, and ARMA models. However, in the last twenty years a lot of more complicated stochastic process models have been introduced, such as, nonlinear time series models, diffusion processes, point processes, and nonergodic processes. This chapter is devoted to providing a modern asymptotic estimation and testing theory for those various stochastic process models. The approach is mainly based on the LAN results given in the previous chapter. More concretely, in Section 3.1 we discuss the asymptotic estimation and testing theory for non-Gaussian vector linear processes in view of LAN. The results are very general and grasp a lot of other works dealing with AR, MA, and ARMA models as special cases. Section 3.2 reviews some elements of nonlinear time series models and the asymptotic estimation theory based on the conditional least squares estimator and maximum likelihood estimator (MLE). We address the problem of statistical model selection in general fashion. Also the asymptotic theory for nonergodic models is mentioned. Recently much attention has been paid to continuous time processes (especially diffusion processes), which appear in finance. Hence, in Section 3.3 we describe the foundation of stochastic integrals and diffusion processes. Then the LAN-based asymptotic theory of estimation for them is studied.
KeywordsPoint Process Maximum Likelihood Estimator Asymptotic Theory Linear Process Final Prediction Error
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