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.
References
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References
- K. S. Vastola and H. V. Poor, 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.
- U. Grenander, A prediction problem in game theory, Ark. Mat. 3 (1957), 371–379. MR 0090486 (19:822g)
- J. Franke, On the robust prediction and interpolation of time series in the presence of correlated noise, J. Time Series Analysis 5 (1984), no. 4, 227–244. MR 782077 (86i:62192)
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- M. P. Moklyachuk, Estimates of stochastic processes from observations with noise, Theory Stoch. Process. 3(19) (1997), no. 3–4, 330–338.
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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
