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Duals of random vectors and processes with applications to prediction problems with missing values

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  • Kasahara, Yukio
  • Pourahmadi, Mohsen
  • Inoue, Akihiko
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    Abstract

    Important results in prediction theory dealing with missing values have been obtained traditionally using difficult techniques based on duality in Hilbert spaces of analytic functions [Nakazi, T., 1984. Two problems in prediction theory. Studia Math. 78, 7-14; Miamee, A.G., Pourahmadi, M., 1988. Best approximations in and prediction problems of Szegö, Kolmogorov, Yaglom, and Nakazi. J. London Math. Soc. 38, 133-145]. We obtain and unify these results using a simple finite-dimensional duality lemma which is essentially an abstraction of a regression property of a multivariate normal random vector (Rao, 1973) or its inverse covariance matrix. The approach reveals the roles of duality and biorthogonality of random vectors in dealing with infinite-dimensional and difficult prediction problems. A novelty of this approach is its reliance on the explicit representation of the prediction error in terms of the data rather than the predictor itself as in the traditional techniques. In particular, we find a new and explicit formula for the dual of the semi-finite process {Xt;t

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 79 (2009)
    Issue (Month): 14 (July)
    Pages: 1637-1646

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    Handle: RePEc:eee:stapro:v:79:y:2009:i:14:p:1637-1646

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    1. Bondon, Pascal, 2002. "Prediction with incomplete past of a stationary process," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 67-76, March.
    2. Pascal Bondon, 2005. "Influence of Missing Values on the Prediction of a Stationary Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(4), pages 519-525, 07.
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    Cited by:
    1. Luati, Alessandra & Proietti, Tommaso & Reale, Marco, 2011. "The Variance Profile," MPRA Paper 30378, University Library of Munich, Germany.

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