In this paper, we consider the problem of the mean square optimal estimation of linear functionals which depend on unknown values of a stationary stochastic sequence based on observations of the sequence with a stationary noise. 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 spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible densities.
The problem of the mean-square optimal estimation of the linear functional AsξFormulas 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 ξ(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 favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.
The problem of mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence from observations of the sequence with a noise and missing observations is considered. Formulas for calculating the meansquare errors and the spectral characteristics of the optimal linear estimates of the functionals are proposed under the condition of spectral certainty, where spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.
The problem of the mean-square optimal estimation of the linear functional Asξ = s−1 ∑ l=0 M l +N l+1 ∑ j=M l a(j)ξ(j), M l = l ∑ k=0 (N k + K k), N 0 = K 0 = 0, which depends on the unknown values of a stochastic stationary sequence ξ(j), j ∈ Z from observations of the sequence at points of time j ∈ Z\S, S = s−1 ∪ l=0 {M l , M l + 1,. .. , 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 ξ(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 favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.
The problem of optimal linear estimation of the functionalwhich depends on unknown values of a stochastic stationary sequence ξ(j) with the help of observations of the sequence ξ(j) + η(j) at points j ∈ Z \ S, where S = s−1 l=0 {M l , . . . , M l + N l+1 }, is considered under the assumption that the sequences {ξ(j)} and {η(j)} are mutually uncorrelated. Formulas for calculating the mean-square error and spectral characteristic of the optimal linear estimator of the functional are proposed under the condition of spectral certainty, where both spectral densities of the sequences ξ(j) and η(j) are known. The minimax (robust) method of estimation is applied in the case where the spectral densities of the sequences ξ(j) and η(j) are not known exactly, but the sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.
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