Abstract
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.
Highlights
The problem of estimation of unknown values of stochastic processes is of great interest both in the theory of stochastic processes and applications of this theory to the data analysis in such fields of science as oceanography, meteorology, astronomy, radio physics, statistical hydromechanics etc
Efficient solution methods of estimation problems of stationary sequences were developed by Kolmogorov
Constructive methods of solution of the estimation problems for stationary stochastic processes were proposed by Wiener [43] and Yaglom [44, 45]
Summary
The problem of estimation of unknown values of stochastic processes is of great interest both in the theory of stochastic processes and applications of this theory to the data analysis in such fields of science as oceanography, meteorology, astronomy, radio physics, statistical hydromechanics etc. To solve the problem in this case one usually finds parametric or nonparametric estimates of the unknown spectral densities and under assumption that the estimated densities are the true ones, one applies the traditional estimation methods This procedure can result in significant increasing of the value of error as Vastola and Poor [42] have demonstrated with the help of some examples. A method of solution of problems of minimax extrapolation and interpolation of stationary sequences which is based on convex optimization methods is proposed in the works by Franke [5], Franke and Poor [6] This approach makes it possible to find equations that determine the least favorable spectral densities for different classes of densities. The problem is investigated in the case of spectral k=0 certainty where the spectral densities of the sequences {ξ(j), j ∈ Z} and {η(j), j ∈ Z} are exactly known and in the case of spectral uncertainty where the spectral densities are not exactly known while a set of admissible spectral densities is given
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