# Extrapolation problem for periodically correlated stochastic sequences with missing observations

## Extrapolation problem for periodically correlated stochastic sequences

## 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

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (2), 39–52. https://doi.org/10.17721/1812-5409.2021/2.6

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