# Chapter 7. Estimation of Multidimensional Stationary Stochastic Sequences from Observations in Special Sets of Points

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Oleksandr Masyutka1 and Mikhail Moklyachuk2
1
Department of Mathematics and Theoretical Radiophysics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
2Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications

### Abstract

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a stochastic stationary sequence is considered. Estimates are based on observations of the sequence with missing data. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimate of the functionals are derived under the condition of spectral certainty, where the spectral density of the sequence is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density of the sequence is not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

Keywords: stationary sequence, mean square error, minimax-robust estimate, least
favourable spectral density, minimax spectral characteristics

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