This work derives new results on strong consistent estimation and prediction for autoregressive processes of order 1 in a separable Banach space B. The consistency results are obtained for the component-wise estimator of the autocorrelation operator in the norm of the space L(B) of bounded linear operators on B. The strong consistency of the associated plug-in predictor then follows in the B-norm. A Gelfand triple is defined through the Hilbert space constructed in Kuelbs' lemma [25]. A Hilbert-Schmidt embedding introduces the Reproducing Kernel Hilbert space (RKHS), generated by the autocovariance operator, into the Hilbert space conforming the Rigged Hilbert space structure. This paper extends the work of Bosq [10] and Labbas and Mourid [26].
SummaryNew results on functional prediction of the Ornstein-Uhlenbeck process in an autoregressive Hilbert-valued and Banach-valued frameworks are derived. Specifically, consistency of the maximum likelihood estimator of the autocorrelation operator, and of the associated plug-in predictor is obtained in both frameworks.
This paper presents new results on prediction of linear processes in function spaces. The autoregressiveHilbertian process framework of order one (ARH(1) process framework) is adopted. A componentwise estimator of the autocorrelation operator is formulated, from the moment-based estimation of its diagonal coefficients, with respect to the orthogonal eigenvectors of the auto-covariance operator, which are assumed to be known.Mean-square convergence to the theoretical autocorrelation operator, in the space of Hilbert-Schmidt operators, is proved. Consistency then follows in that space. For the associated ARH(1) plug-in predictor, mean absolute convergence to the corresponding conditional expectation, in the considered Hilbert space, is obtained. Hence, consistency in that space also holds. A simulation study is undertaken to illustrate the finite-large sample behavior of the formulated componentwise estimator and predictor. The performance of the presented approach is compared with alternative approaches in the previous and current ARH (1) framework literature, including the case of unknown eigenvectors. mean absolute and quadratic convergence. Cuevas [2014]; Horváth and Kokoszka [2012]; Hsing and Eubank [2015], and in a recent Special Issue of this journal Goia and Vieu [2016]. These references include a nice summary on the statistics theory for functional data, contemplating covariance operator theory and eigenfunction expansion, perturbation theory, smoothing and regularization, probability measures on a Hilbert spaces, functional principal component analysis, functional counterparts of the multivariate canonical correlation analysis, the two sample problem and the change point problem, functional linear models, functional test for independence, functional time series theory, spatially distributed curves, software packages and numerical implementation of the statistical procedures discussed, among other topics. The special case of functional regression models, in which the predictor is a random function and the response is scalar, has been particularly well studied. Various specifications of the functional regression parameter arise in fields such as biology, climatology, chemometrics, and economics. To avoid the computational (high-dimensional) limitations of the nonparametric approach, several parametric and semi-parametric methods have been proposed; see, e.g., Ferraty et al. [2012] and the references therein. In Ferraty et al. [2012], a combination of a spline approximation and the one-dimensional Nadaraya-Watson approach was proposed to avoid high dimensionality issues. Generalizations to the case of more regressors (all functional, or both functional and real) were also addressed in the nonparametric, semi-parametric, and parametric frameworks; for an overview, see Aneiros-Pérez and Vieu [2006]; Febrero-Bande and González-Manteiga [2013]; Ferraty and Vieu [2009]. In the nonparametric regression framework, the case where the covariates and the response are functional was considered by Ferraty et al. [2012],...
This work adopts a Banach-valued time series framework for estimation and prediction, from temporal correlated functional data, in presence of exogenous variables. The consistency of the proposed functional predictor is illustrated in the simulation study undertaken. Air pollutants PM 10 curve forecasting, in the Haute-Normandie region (France), is addressed, by implementation of the functional time series approach presented.Keywords Air pollutants forecasting • Banach spaces • functional time series • meteorological variables • strong consistency
New results on strong-consistency in the trace operator norm are obtained, in the parameter estimation of an autoregressive Hilbertian process of order one (ARH(1) process). Additionally, a strongly-consistent diagonal componentwise estimator of the autocorrelation operator is derived, based on its empirical singular value decomposition.
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