Abstract
The autoregressive Hilbertian process framework has been introduced in Bosq (2000). This book provides the nonparametric estimation of the autocorrelation and covariance operators of the autoregressive Hilbertian processes. The asymptotic properties of these estimators are also provided. The maximum likelihood approach still remains unexplored. This paper obtains the asymptotic distribution of the maximum likelihood (ML) estimators of the auto-covariance operator of the Hilbert-valued innovation process, and of the autocorrelation operator of a Gaussian autoregressive Hilbertian process of order one. A real data example is analyzed in the financial context for illustration of the performance of the projection maximum likelihood estimation methodology in the context of missing data.
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References
Aalen OO, Gunnes N (2010) A dynamic approach for reconstructing missing longitudinal data using the linear increments model. Biostatistics 11:453–472
Basse M, Diop A, Dabo-Niang S (2008) Mean square properties of a class of kernel density estimates for spatial functional random variables. In: Annales De L’I.S.U.P. Publications de l’Institut de Statistique de l’Université de Paris. Numéro Spécial-Volume LII. Fascicule 1–2, Paris, pp 91–108
Bosq D (2000) Linear processes in function spaces. Springer-Verlag
Bosq D, Blanke D (2007) Inference and predictions in large dimensions. Wiley, New York
Da Prato G, Zabczyk J (2002) Second order partial differential equations in Hilbert spaces. Cambridge University Press, New York
Dautray R, Lions J-L- (1985) Mathematical analysis and numerical methods for science and technology, vol 3: spectral theory and applications. Springer. ISBN 978-3-540-66099-6 (2nd printing, 2000, X, p 542)
Ferraty F, Vieu P (2006) Nonparameric functional data analysis. Springer, New York
Guillas S, Lai M-J (2010) Bivariate splines for spatial functional regression models. J Nonparametr Stat 22:477–497
Hörmann S, Kokoszka P (2010) Weakly dependent functional data. Ann Stat 38:1845–1884
Nakai M, Ke W (2011) Review of methods for handling missing data in longitudinal data analysis. Int J Math Anal 5:1–13
Ramsay J, Silverman B (2005) Functional data analysis. Springer, New York
Ruiz-Medina MD (2011) Spatial autoregressive and moving average Hilbertian processes. J Multivar Anal 102:292–305
Ruiz-Medina MD (2012) Spatial functional prediction from spatial autoregressive Hilbertian processes. Environmetrics 23:119–128
Ruiz-Medina MD, Salmerón R (2010) Functional maximum-likelihood estimation of ARH(p) models. Stoch Environ Res Risk Assess 24:131–146
Ruiz-Medina MD, Salmerón R (2011) Asymptotic properties of functional maximum-likelihood ARH parameter estimators. Rev Nouv Technol Inf 33–58 (Special issue of JSFdS Bordeaux 2009)
Salmerón R, Ruiz-Medina MD (2009) Multispectral decomposition of functional autoregressive models. Stoch Environ Res Risk Assess 23:289–297
Walk H (1977) An invariance principle for the Robbins–Monro process in a Hilbert space. Z Wahrscheinlichkeitstheor Verw Geb 39:135–150
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Ruiz-Medina, M.D., Espejo, R.M. Maximum-Likelihood Asymptotic Inference for Autoregressive Hilbertian Processes. Methodol Comput Appl Probab 17, 207–222 (2015). https://doi.org/10.1007/s11009-013-9329-8
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DOI: https://doi.org/10.1007/s11009-013-9329-8
Keywords
- Autoregressive Hilbertian processes
- Central limit results
- EM algorithm
- Financial data
- Maximum likelihood estimation
- Missing functional data
- Numerical projection methods