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

ARH(1) plug-in,
Asymptotic properties,
Predictor’s prowess.
Dimension reduction
Functional data
Functional time series

Javier Álvarez Liébana, Denis Bosq, María Dolores Ruiz Medina, «Asymptotic properties of a component-wise ARH(1) plug-in predictor», J. Multivariate Anal. 155, 12–34 (2017), doi: 10.1016/j.jmva.2016.11.009


Universidad Complutense de Madrid

Denis Bosq

Universitè Pierre et Marie Curie

Universidad de Granada


March 2017

Otros detalles

Artículo derivado de mi estancia en París con Denis Bosq



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.

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Cita BibTeX

  author = {J. Álvarez-Liébana and D. Bosq and M. D. Ruiz-Medina},
  title = {Asymptotic properties of a component-wise ARH(1) plug-in predictor},
  journal = {J. Multivariate Anal.},
  volume = {155},
  pages = {12-34},
  keywords = {ARH(1) processes, consistency, functional prediction, mean absolute and quadratic convergence},
  url = {},
  year = {2017}