Abstract
In order to predict a continuous time process on an entire time-interval, we introduce and study a Hilbertian autoregressive process Xt, t=0, ±1, ±2, …
First, we present consistent nonparametric estimators for the covariance operator C of Xt and the cross estimator covariance operator of (Xt, Xt+1). Then we construct an estimator pn of the autocorrelation operator, by projecting the data on a finite dimensional subspace vk n, generated by eigenvectors of C. If the eigenvectors are unknown, we construct a preliminary estimator of vk n. Under mild regularity conditions, we show that the predictor based upon pn converges in probability. When the process (Xt) satisfies an additional mixing condition, we obtain almost sure convergence.
Rates of convergence are given as well as numerical examples.
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© 1991 Springer Science+Business Media Dordrecht
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Bosq, D. (1991). Modelization, Nonparametric Estimation and Prediction for Continuous Time Processes. In: Roussas, G. (eds) Nonparametric Functional Estimation and Related Topics. NATO ASI Series, vol 335. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3222-0_38
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DOI: https://doi.org/10.1007/978-94-011-3222-0_38
Publisher Name: Springer, Dordrecht
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