Interpolation Problem for Stationary Sequences with Missing Observations

  • Mikhail Moklyachuk Kyiv National Taras Shevchenko University
  • Maria Sidei Kyiv National Taras Shevchenko University
Keywords: Stationary sequence, mean square error, minimax-robust estimate, least favorable spectral density, minimax spectral characteristic

Abstract

The problem of the mean-square optimal estimation of the linear functional $A_s\xi=\sum\limits_{l=0}^{s-1}\sum\limits_{j=M_l}^{M_l+N_{l+1}}a(j)\xi(j),$ $M_l=\sum\limits_{k=0}^l (N_k+K_k),$ \, $N_0=K_0=0,$ which depends on the unknown values of a stochastic stationary sequence $\xi(k)$ from observations of the sequence at points of time $j\in\mathbb{Z}\backslash S $, $S=\bigcup\limits_{l=0}^{s-1}\{ M_{l}, M_{l}+1, \ldots, M_{l}+N_{l+1} \}$ is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty, where the spectral density of the sequence $\xi(j)$ is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density is not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

Author Biography

Mikhail Moklyachuk, Kyiv National Taras Shevchenko University
Department of Probability Theory, Statistics and Actuarial Mathematics, Professor

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Published
2015-08-28
How to Cite
Moklyachuk, M., & Sidei, M. (2015). Interpolation Problem for Stationary Sequences with Missing Observations. Statistics, Optimization & Information Computing, 3(3), 259-275. https://doi.org/10.19139/soic.v3i3.149
Section
Research Articles