On the problem of filtration for vector stationary sequences

Authors:
M. P. Moklyachuk and O. Yu. Masyutka

Translated by:
V. V. Semenov

Journal:
Theor. Probability and Math. Statist. **75** (2007), 109-119

MSC (2000):
Primary 60G35; Secondary 62M20, 93E10, 93E11

DOI:
https://doi.org/10.1090/S0094-9000-08-00718-7

Published electronically:
January 24, 2008

MathSciNet review:
2321185

Full-text PDF Free Access

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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.

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References
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*An analysis of the effects of spectral uncertainty on Wiener filtering*, Automatica **28** (1983), 289–293.
- S. A. Kassam and H. V. Poor,
*Robust techniques for signal processing: A survey*, Proc. IEEE **73** (1985), no. 3, 433–481.
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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

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