Asymptotic theory of statistical inference for time series
The goal of the book is to provide modern statistical techniques and theory for stochastic processes. The presentation is based mainly on the recent results of the authors on asymptotic statistics, which appear for the first time in a monograph. The authors deal with a wide variety of stochastic processes, such as non-Gaussian linear processes, long-memory processes, nonlinear processes, orthogonal increment processes, and continuous time processes. They develop estimation and testing theory, and also discriminant analysis, nonparametric methods, higher order asymptotic theory, large deviation principles, and saddlepoint approximations. The key concept of the book is local asymptotic normality (LAN), which gives a unified view for the time series asymptotic theory. Throughout the book, numerical examples to real data are given as applications. Chapter 1 reviews the elements of stochastic processes. Chapter 2 describes a general LAN theorem, which is applied to a wide class of vector valued processes. Chapter 3 discusses the asymptotic efficiency of estimators and tests for nonlinear models, non-ergodic models, and diffusion processes. In Chapter 4, Edgeworth expansions are used to derive higher order asymptotics (h.o.a.) of estimators and tests. This is done in view of statistical differential geometry. Basing on an infinite dimensional differential calculus of Wiener functionals, the h.o.a. are considered for diffusion processes. The h.o.a. are developed for transformations of statistics and iterative methods, e.g. the Newton-Raphson method. In Chapter 5 a modern asymptotic estimation and testing theory for long-memory processes is studied. These processes are very fashionable because of plausible applications in economics and financial modelling. The asymptotic theory for regression models with long-memory disturbances shows different features from that for short-memory processes. In Chapter 6 statistical analysis based on (nonlinear) integral functionals of spectal densities is considered. Robust estimation and testing are studied, where robustness is understood in the sense that the statistical procedures are not influenced by the presence of a decaying trend or periodic components. Discriminant analysis for stationary time series is treated in Chapter 7. The standard methods for Gaussian processes are used, and then a modern freuqency domain approach is introduced. The problem for discriminating spectra of linear processes is discussed, both by parametric and nonparametric approaches. Chapter 8 deals with classical large deviation theory and a modern saddlepoint approximation method, which performs better when the sample size is small. Numerical applications to ratio statistics of quadratic forms of normal variables are given. Each chapter finishes with a list of problems. Some of them are elementary, the others present related results supplemented with references. This monograph is suitable as a good reference book and a handbook on the statistical analysis of stochastic processes. It is not a textbook, because the exposition is very concentrated, and the results are presented in great generality. The book will be useful to researchers in econometrics, financial mathematics, seismology, and mathematical statistics.
scientific article; zbMATH DE number 1517514