{"ID":5937191,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-09T08:41:17.711438627Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.04823","arxiv_id":"2607.04823","title":"Minimax prediction in the Functional Autoregressive Model","abstract":"The Functional Autoregressive Model (FAR) generalizes the multivariate AR(1) model in Time Series Analysis to functional data. It serves as a historical foundational point in the study of functional time series and remains a fundamental and widely used model for dependent functional data. The process-the observed data-generated by the FAR model forms a Hilbert-valued Markov chain. This paper investigates the non-asymptotic prediction mean square error and derives a lower bound. This lower bound is established in the specific context of non-i.i.d. data and depends on the mixed smoothness of the functional time series and of the unknown correlation operator driving the FAR model. Instead of the standard functional PCA regularization, a ridge-type estimator is proposed, which avoids the preliminary estimation of the spectrum of the covariance sequence associated with the process. A non-asymptotic upper bound is derived for this estimate, which matches the lower bound up to multiplicative constants. Furthermore, a detailed study of the estimate's bias reveals connections between functional smoothness parameters and regularly/rapidly varying functions, which are common in extreme value theory. Simulation results corroborate the theoretical main theorems.","short_abstract":"The Functional Autoregressive Model (FAR) generalizes the multivariate AR(1) model in Time Series Analysis to functional data. It serves as a historical foundational point in the study of functional time series and remains a fundamental and widely used model for dependent functional data. The process-the observed data-...","url_abs":"https://arxiv.org/abs/2607.04823","url_pdf":"https://arxiv.org/pdf/2607.04823v1","authors":"[\"André Mas\",\"Angelina Roche\"]","published":"2026-07-06T08:53:58Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}