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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

On the problem of filtration for vector stationary sequences


Authors: M. P. Moklyachuk and O. Yu. Masyutka
Translated by: V. V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal: Theor. Probability and Math. Statist. 75 (2007), 109-119
MSC (2000): Primary 60G35; Secondary 62M20, 93E10, 93E11
Published electronically: January 24, 2008
MathSciNet review: 2321185
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the problem of optimal linear estimation of the functional $ A\vec \xi=\sum_{j=0}^\infty {\vec a(j)\vec \xi ( - j)}$ depending on unknown values of a vector stationary sequence $ \vec \xi (j)=\{\xi_k(j)\}_{k=1}^T $ from observations upon the sequence $ \vec \xi (j)+\vec \eta (j)$ for $ j \leq 0$ where $ \vec \eta (j)=\{\eta_k(j)\}_{k=1}^T $ is a vector stationary sequence, being uncorrelated with $ \vec \xi (j)$. We obtain relations for the mean square error and spectral characteristic of the optimal estimator of the functional. We also find the least favorable spectral densities and minimax (robust) spectral characteristics of optimal estimators of the functional for a particular class $ D$ of spectral densities.


References [Enhancements On Off] (What's this?)

  • 1. K. S. Vastola and H. V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica 28 (1983), 289-293.
  • 2. S. A. Kassam and H. V. Poor, Robust techniques for signal processing: A survey, Proc. IEEE 73 (1985), no. 3, 433-481.
  • 3. Ulf Grenander, A prediction problem in game theory, Ark Mat. 3 (1957), 371–379. MR 0090486 (19,822g)
  • 4. Jürgen Franke, On the robust prediction and interpolation of time series in the presence of correlated noise, J. Time Ser. Anal. 5 (1984), no. 4, 227–244. MR 782077 (86i:62192), http://dx.doi.org/10.1111/j.1467-9892.1984.tb00389.x
  • 5. Jürgen Franke, Minimax-robust prediction of discrete time series, Z. Wahrsch. Verw. Gebiete 68 (1985), no. 3, 337–364. MR 771471 (86f:62164), http://dx.doi.org/10.1007/BF00532645
  • 6. M. P. Moklyachuk, Estimates of stochastic processes from observations with noise, Theory Stoch. Process. 3(19) (1997), no. 3-4, 330-338.
  • 7. M. P. Moklyachuk, Robust procedures in time series analysis, Theory Stoch. Process. 6(22) (2000), no. 3-4, 127-147.
  • 8. M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory Stoch. Process. 7(23) (2001), no. 1-2, 253-264.
  • 9. Yu. A. Rozanov, Statsionarnye sluchainye protsessy, 2nd ed., \cyr Teoriya Veroyatnosteĭ\ i Matematicheskaya Statistika [Probability Theory and Mathematical Statistics], vol. 42, “Nauka”, Moscow, 1990 (Russian). MR 1090826 (92d:60046)

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Additional Information

M. P. Moklyachuk
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email: mmp@univ.kiev.ua

O. Yu. Masyutka
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine

DOI: http://dx.doi.org/10.1090/S0094-9000-08-00718-7
PII: S 0094-9000(08)00718-7
Keywords: Vector stationary sequence, observations in the presence of noise, optimal linear estimator, mean square error, spectral characteristic, least favorable spectral density, minimax (robust) spectral characteristic
Received by editor(s): January 20, 2006
Published electronically: January 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society