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Theory of Probability and Mathematical Statistics

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Extrapolation of periodically correlated stochastic processes observed with noise


Authors: I. I. Dubovets’ka and M. P. Moklyachuk
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 88 (2013).
Journal: Theor. Probability and Math. Statist. 88 (2014), 67-83
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI: https://doi.org/10.1090/S0094-9000-2014-00919-9
Published electronically: July 24, 2014
MathSciNet review: 3112635
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of the optimal linear estimation of the functional

$\displaystyle A\zeta =\int _0^\infty a(t)\zeta (t)\,dt $

depending on unknown values of a periodically correlated stochastic process $ \zeta (t)$. An estimator is constructed from observations of the process $ \zeta (t)+\theta (t)$ for $ t<0$, where $ \theta (t)$ is a periodically correlated process being uncorrelated with $ \zeta (t)$. Formulas for calculating the spectral characteristic and the mean square error of the optimal linear estimator of the functional are proposed in the case where spectral densities are known. In the case where spectral densities are not known but a set of admissible spectral densities is specified, formulas that determine the least favorable spectral density and the minimax (robust) spectral characteristics of optimal estimators of the above functional are proposed.

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Additional Information

I. I. Dubovets’ka
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Kyiv National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: idubovetska@gmail.com

M. P. Moklyachuk
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Kyiv National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: mmp@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2014-00919-9
Keywords: Periodically correlated processes, minimax estimator, mean square error, least favorable spectral density
Received by editor(s): October 4, 2012
Published electronically: July 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society