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J. Álvarez-Liébana

Publications and source records attributed to J. Álvarez-Liébana.

8 recordsLinked to original sources
arXiv2022

A goodness-of-fit test for functional time series with applications to Ornstein-Uhlenbeck processes

High-frequency financial data can be collected as a sequence of curves over time; for example, as intra-day price, currently one of the topics of greatest interest in finance. The Functional Data Analysis framework provides a suitable tool to extract the information contained in the shape of the daily paths, often unavailable from classical statistical methods. In this paper, a novel goodness-of-fit test for autoregressive Hilbertian (ARH) models, with unknown and general order, is proposed. The test imposes just the Hilbert-Schmidt assumption on the functional form of the autocorrelation operator, and the test statistic is formulated in terms of a Cramér-von Mises norm. A wild bootstrap resampling procedure is used for calibration, such that the finite sample behavior of the test, regarding power and size, is checked via a simulation study. Furthermore, we also provide a new specification test for diffusion models, such as Ornstein-Uhlenbeck processes, illustrated with an application to intra-day currency exchange rates. In particular, a two-stage methodology is proffered: firstly, we check if functional samples and their past values are related via ARH(1) model; secondly, under linearity, we perform a functional F-test.

stat.ME↗
arXiv2018

Strongly consistent autoregressive predictors in abstract Banach spaces

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 componentwise estimator of the autocorrelation operator in the norm of the space $\mathcal{L}(B)$ 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 \cite{Kuelbs70}. 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 \cite{Bosq00} and \cite{LabbasMourid02}.

math.ST↗
arXiv2018

Classical and bayesian componentwise predictors for non-compact correlated ARH(1) processes

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.

stat.AP↗
arXiv2018

Asymptotic properties of a componentwise ARH(1) plug-in predictor

This paper presents new results on prediction of linear processes in function spaces. The autoregressive Hilbertian 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.

math.ST↗
arXiv2018

The effect of the spatial domain in FANOVA models with ARH(1) error term

Functional Analysis of Variance (FANOVA) from Hilbert-valued 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, extending the formulation given in Ruiz-Medina (2016). 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.

stat.AP↗
arXiv2017

A note on strong-consistency of componentwise ARH(1) predictors

This paper presents a new result on strong-consistency, in the trace norm, of a diagonal componentwise parameter estimator of the autocorrelation operator of an autoregressive process of order one (ARH(1) process), allowing strong-consistency of the associated plug-in predictor. These results are derived, when the eigenvectors of the autocovariance operator are unknown, and the autocorrelation operator does not admit a diagonal spectral representation with respect to the eigenvectors of the autocovariance operator.

math.ST↗
arXiv2017

A review and comparative study on functional time series techniques

This paper reviews the main estimation and prediction results derived in the context of functional time series, when Hilbert and Banach spaces are considered, specially, in the context of autoregressive processes of order one (ARH(1) and ARB(1) processes, for H and B being a Hilbert and Banach space, respectively). Particularly, we pay attention to the estimation and prediction results, and statistical tests, derived in both parametric and non-parametric frameworks. A comparative study between different ARH(1) prediction approaches is developed in the simulation study undertaken.

math.ST↗