The functional linear model with functional response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this article, we propose a novel goodness-of-fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cramér–von Mises norm over a doubly projected empirical process which, using geometrical arguments, yields an easy-to-compute weighted quadratic norm. A resampling procedure calibrates the test through a wild bootstrap on the residuals and the use convenient computational procedures. As a sideways contribution, and since the statistic requires a reliable estimator of the FLMFR, we discuss and compare several regularized estimators, providing a new one specifically convenient for our test. The finite sample behavior of the test is illustrated via a simulation study. Also, the new proposal is compared with previous significance tests. Two novel real data sets illustrate the application of the new test.

This work adopts a Banach-valued time series framework for component-wise estimation and prediction, from temporal correlated functional data, in presence of exogenous variables. The strong-consistency of the proposed functional estimator and associated plug-in predictor is formulated. The simulation study undertaken illustrates their large-sample size properties. Air pollutants PM10 curve forecasting, in the Haute-Normandie region (France), is addressed by implementation of the functional time series approach presented.

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.

Esta tesis proporciona nuevos resultados en el contexto de la estimación y predicción funcional, a partir de modelos autorregresivos Hilbertianos, o bien, con valores en espacios de Banach separables. El objetivo fundamental es proporcionar herramientas adecuadas para modelizar relaciones lineales entre variables aleatorias funcionales, que dependen de un índice temporal.

Functional Analysis of Variance (FANOVA) from Hilbertvalued correlated data with spatial rectangular or circular supports is analyzed, when Dirichlet conditions are assumed on the boundary. Specifically, a Hilbert-valued fixed effect model with error term defined from an Autoregressive Hilbertian process of order one (ARH(1) process) is considered. A new statistical test is also derived to contrast the significance of the functional fixed effect parameters. The Dirichlet conditions established at the boundary affect the dependence range of the correlated error term. While the rate of convergence to zero of the eigenvalues of the covariance kernels, characterizing the Gaussian functional error components, directly affects the stability of the generalized least-squares parameter estimation problem. A simulation study and a real-data application related to fMRI analysis are undertaken to illustrate the performance of the parameter estimator and statistical test derived.

A special class of standard Gaussian Autoregressive Hilbertian processes of order one (Gaussian ARH(1) processes), with bounded linear autocorrelation operator, which does not satisfy the usual Hilbert–Schmidt assumption, is considered. To compensate the slow decay of the diagonal coefficients of the autocorrelation operator, a faster decay velocity of the eigenvalues of the trace autocovariance operator of the innovation process is assumed. As usual, the eigenvectors of the autocovariance operator of the ARH(1) process are considered for projection, since, here, they are assumed to be known. Diagonal componentwise classical and bayesian estimation of the autocorrelation operator is studied for prediction. The asymptotic efficiency and equivalence of both estimators is proved, as well as of their associated componentwise ARH(1) plugin predictors. A simulation study is undertaken to illustrate the theoretical results derived.

This paper presents new results on the prediction of linear processes in function spaces. The autoregressive Hilbertian process framework of order one (ARH(1) framework) is adopted. A component-wise estimator of the autocorrelation operator is derived from the moment-based estimation of its diagonal coefficients with respect to the orthogonal eigenvectors of the autocovariance operator, which are assumed to be known. Mean-square convergence to the theoretical autocorrelation operator is proved in the space of Hilbert–Schmidt operators. Consistency then follows in that space. Mean absolute convergence, in the underlying Hilbert space, of the ARH(1) plug-in predictor to the conditional expectation is obtained as well. A simulation study is undertaken to illustrate the large-sample behavior of the formulated component-wise estimator and predictor. Additionally, alternative component-wise (with known and unknown eigenvectors), regularized, wavelet-based penalized, and nonparametric kernel estimators of the autocorrelation operator are compared with the one presented here, in terms of prediction.

New 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.

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