Gaussian and non-Gaussian linear time series and random fields
This book is devoted mainly to AR and ARMA linear stationary sequences and random fields. If the sequence is Gaussian then it is reversible and best predictors are linear. However, the best predictors may be non-linear in the non-Gaussian case. The author presents necessary and sufficient conditions for reversibility. If \(\{\xi_t\}\) is a white noise and an ARMA process \(\{x_t\}\) is given by \(\phi(B)x_t=\theta(B)\xi_t\) (in usual notation) then \(\{x_t\}\) is said to be minimum phase if the polynomials \(\phi(z)\), \(\theta(z)\) have their zeros outside the closed unit disc in the complex plane. The parameter estimation is first discussed in the Gaussian case. It is shown that the same estimates (called quasi-Gaussian estimates) can also be used in the minimum phase non-Gaussian case but they are not asymptotically optimal. The material concerning homogeneous random fields is focused on the interpolation problem. The classical Whittle's approximation and tapering are used for estimating parameters. In a non-Gaussian one-dimensional minimum phase series \(\{x_t\}\) the best predictor is still linear. However, if \(\{x_t\}\) is nonminimum phase then the best predictor is generally nonlinear. The author presents some of his results which concern this topic. Further, quasi-Gaussian likelihood is investigated and the moderate sample behavior of the estimates is derived. Markov fields and Markov chains of a possibly non-Gaussian character are discussed. Some asymptotic results are given for the estimation procedure which uses approximate maximum likelihood for nonminimum phase AR and ARMA processes. Notes on the history of the presented results are given at the end of the book. This publication can be recommended to readers familiar with the basic concepts of time series who are interested in estimation problems in nonminimum phase processes.
scientific article; zbMATH DE number 1402080