Interpolation Problem for Stationary Sequences with Missing Observations
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.References
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