Extrapolation problem for periodically correlated stochastic sequences with missing observations: Extrapolation problem for periodically correlated stochastic sequences

I. I. Golichenko; O. Yu. Masyutka; Mykhajlo Pavlovych Moklyachuk

doi:10.17721/1812-5409.2021/2.6

Authors

I. I. Golichenko National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37, Prosp. Peremohy, Kyiv, Ukraine, 03056 https://orcid.org/0000-0001-5639-8271
O. Yu. Masyutka Taras Shevchenko National University of Kyiv https://orcid.org/0000-0002-7301-8813
Mykhajlo Pavlovych Moklyachuk Taras Shevchenko National University of Kyiv https://orcid.org/0000-0002-6173-0280

DOI:

https://doi.org/10.17721/1812-5409.2021/2.6

Keywords:

periodically correlated sequence, optimal linear estimate, mean square error, least favourable spectral density matrix, minimax spectral characteristic

Abstract

The problem of optimal estimation of the linear functionals $A{\zeta}=\sum_{j=1}^{\infty}{a}(j){\zeta}(j),$
which depend on the unknown values of a periodically correlated stochastic sequence ${\zeta}(j)$ from observations of the sequence ${\zeta}(j)+{\theta}(j)$ at points $j\in\{...,-n,...,-2,-1,0\}\setminus S$, $S=\bigcup _{l=1}^{s-1}\{-M_l\cdot T+1,\dots,-M_{l-1}\cdot T-N_{l}\cdot T\}$, is considered, where ${\theta}(j)$ is an uncorrelated with ${\zeta}(j)$ periodically correlated stochastic sequence. Formulas for calculation the mean square error and the spectral characteristic of the optimal estimate of the functional $A\zeta$ are proposed in the case where spectral densities of the sequences are exactly known. Formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of functionals are proposed in the case of spectral uncertainty, where the spectral densities are not exactly known while some sets of admissible spectral densities are specified.

Pages of the article in the issue: 39 - 52

Language of the article: English

Author Biography

Mykhajlo Pavlovych Moklyachuk, Taras Shevchenko National University of Kyiv

Професор кафедри теорії ймовірностей, статистики і актуарної математики

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