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On consistent nonparametric order determination and chaos. (English) Zbl 0782.62081
Summary: We give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models. We argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos. Setting the stochastic models within the framework of nonlinear autoregression, we introduce the notion of a generalized partial autocorrelation and an order. We approach the estimation of the embedding dimension via order determination of an unknown nonlinear autoregression by cross-validation, and give justification by proving its consistency under global boundedness.
As a by-product, we provide a theoretical justification of the final prediction error approach of B. Auestad and D. Tøstheim [Biometrika 77, 669-688 (1990)]. Some illustrations based on the Hénon map and several real data sets are given. The bias of the residual sum of squares as essentially a noise variance estimator is quantified.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G07 Density estimation
37N99 Applications of dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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