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Theory of Probability and Mathematical Statistics

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Interpolation of stationary sequences observed with a noise


Authors: M. P. Moklyachuk and M. I. Sidei
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 93 (2015).
Journal: Theor. Probability and Math. Statist. 93 (2016), 153-167
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI: https://doi.org/10.1090/tpms/1000
Published electronically: February 7, 2017
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Abstract: The problem of optimal linear estimation of the functional

$\displaystyle A_s\xi =\sum _{l=0}^{s-1}\sum _{j=M_l}^{M_l+N_{l+1}}a(j)\xi (j), \qquad M_l=\sum _{k=0}^l (N_k+K_k), \qquad N_0=K_0=0, $

which depends on unknown values of a stochastic stationary sequence $ \xi (j)$ with the help of observations of the sequence $ \xi (j)+\eta (j)$ at points $ j\in \mathbb{Z}\setminus S $, where $ S=\bigcup _{l=0}^{s-1}\{ M_{l}, \dots , M_{l}+N_{l+1} \}$, is considered under the assumption that the sequences $ \{\xi (j)\}$ and $ \{\eta (j)\}$ are mutually uncorrelated. Formulas for calculating the mean-square error and spectral characteristic of the optimal linear estimator of the functional are proposed under the condition of spectral certainty, where both spectral densities of the sequences $ \xi (j)$ and $ \eta (j)$ are known. The minimax (robust) method of estimation is applied in the case where the spectral densities of the sequences $ \xi (j)$ and $ \eta (j)$ are not known exactly, but the sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.

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Additional Information

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

M. I. Sidei
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email: marysidei4@gmail.com

DOI: https://doi.org/10.1090/tpms/1000
Keywords: Stationary sequence, robust estimator, mean square error, least favorable spectral density, minimax spectral characteristics
Received by editor(s): October 28, 2015
Published electronically: February 7, 2017
Additional Notes: This paper was prepared following the talk at the International conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright: © Copyright 2017 American Mathematical Society

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