Inference on linear processes in Hilbert and Banach spaces. Statistical analysis of high-dimensional data

Abstract

This PhD thesis focuses on statistical estimation and prediction from temporal correlated functional data. We adopt the functional time series framework, considering, in particular, autoregressive processes in Hilbert and Banach spaces (ARH(1) and ARB(1) processes). Our primary objective is the statistical estimation of the conditional mean, from temporal correlated data, considering linear models in a parametric framework. That is the case, for example, of the estimation of the functional response in linear regression, with functional regressors and correlated errors, lying in Hilbert or Banach spaces. Some extensions to the Bayesian framework are derived as well. Nonparametric classification is also considered, in the special case of spatially supported uncorrelated functional data. Specifically, the main contributions of this PhD thesis can be summarized as follows: • The derivation of new weak- and strong- consistency results, for componentwise estimators of the autocorrelation operator of an ARH(1) process, in the norms of bounded linear, Hilbert-Schmidt and trace operators. Under the same setting of conditions, consistency of the corresponding plug-in predictors is derived as well. The cases of known and unknown eigenvectors are studied. Some particular examples are also analysed, such as the Ornstein-Uhlenbeck process in Hilbert and Banach spaces, as motivation of the subject summarized in the next paragraph. • The extension of the results previously derived on functional prediction, based on ARC(1) and \linebreak ARD(1) processes, with respective values in the space of continuous functions and in the Skorokhod space, to the case of an abstract separable Banach space. Specifically, sufficient conditions are obtained for the strong-consistency of the componentwise estimator of the autocorrelation operator, and the associated plug-in predictor. The methodological approach proposed, in the derivation of these results, is based on the construction appearing in Lemma 2.1 in Kuelbs (1970), and the definition of continuous embeddings between suitable Banach and Hilbert spaces. • The introduction of the Bayesian statistical perspective, in the componentwise estimation of the autocorrelation operator of an ARH(1) process, with the consideration of the corresponding ARH(1) plug-in predictor, under weaker setting of conditions than before for its asymptotic efficiency. The asymptotic equivalence of both, the classical and Bayesian estimators and plug-in predictors, is studied as well. • The FANOVA analysis of functional fixed effect models in Hilbert spaces, under correlated errors, having values in a separable Hilbert space. In this context, non-separable point spectrum matrix covariance operator models are analysed. • A wide range of simulation studies have been undertaken, for comparative purposes, in relation to the existing functional prediction methodologies in the ARH(1), ARB(1) and nonparametric frameworks. • Some real-data applications are considered to illustrate the implementation of the proposed functional estimation and prediction methodologies in practice.