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Filtration of linear functionals of periodically correlated sequences


Authors: I. I. Dubovets′ka and M. P. Moklyachuk
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 86 (2012).
Journal: Theor. Probability and Math. Statist. 86 (2013), 51-64
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI: https://doi.org/10.1090/S0094-9000-2013-00888-6
Published electronically: August 20, 2013
MathSciNet review: 2986449
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Abstract: The problem of the optimal estimation is considered for the linear functional

$\displaystyle A{\zeta }=\sum _{j=0}^\infty {a}(j){\zeta }(-j) $

that depends on unknown values of a periodically correlated stochastic sequence $ \zeta (j)$; the estimator is constructed from observations of the sequence $ \zeta (j)+\theta (j)$, $ j\leq 0$, where $ \theta (j)$ is a periodically correlated noise. We obtain the mean square error and spectral characteristic of the optimal linear estimate of the functional $ A{\zeta }$ in the case where the spectral densities of the sequences that generate $ \zeta (j)$ and $ \theta (j)$ are known. For the case where these spectral densities are unknown but a set of admissible spectral densities is given, we find the least favorable spectral density and minimax spectral characteristic for the optimal estimate of the functional $ A{\zeta }$.

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Additional Information

I. I. Dubovets′ka
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine
Email: idubovetska@gmail.com

M. P. Moklyachuk
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine
Email: mmp@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2013-00888-6
Keywords: Periodically correlated sequence, robust estimate, mean square error, least favorable spectral density, minimax spectral characteristic
Received by editor(s): November 21, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society