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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

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

Suggested Citation

  • Kasahara, Yukio & Pourahmadi, Mohsen & Inoue, Akihiko, 2009. "Duals of random vectors and processes with applications to prediction problems with missing values," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1637-1646, July.
  • Handle: RePEc:eee:stapro:v:79:y:2009:i:14:p:1637-1646
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    References listed on IDEAS

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    1. Mohsen Pourahmadi & E. S. Soofi, 2000. "Prediction Variance and Information Worth of Observations in Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(4), pages 413-434, July.
    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, July.
    3. S. R. Brubacher & G. Tunnicliffe Wilson, 1976. "Interpolating Time Series with Application to the Estimation of Holiday Effects on Electricity Demand," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 25(2), pages 107-116, June.
    4. Bondon, Pascal, 2002. "Prediction with incomplete past of a stationary process," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 67-76, March.
    5. Mohsen Pourahmadi, 1989. "Estimation And Interpolation Of Missing Values Of A Stationary Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(2), pages 149-169, March.
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    1. Cheng, Raymond, 2015. "Prediction of stationary Gaussian random fields with incomplete quarterplane past," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 245-258.
    2. Kohli, P. & Pourahmadi, M., 2014. "Some prediction problems for stationary random fields with quarter-plane past," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 112-125.
    3. Alessandra Luati & Tommaso Proietti & Marco Reale, 2012. "The Variance Profile," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 607-621, June.

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