Interpolation problem for multidimensional harmonizable stable sequences

Анотація

The problem of estimation of unobserved values of stochastic processes is of constant interest in the theory and applications of stochastic processes. The problem of forecasting future values of economic and physical processes, the problem of restoring lost information, cleaning signals, or other data from observations with noise is magnified in an information-laden world. Therefore, the development of estimation methods is one of the main tasks of the modern theory of stochastic processes. In this paper, we consider the problem of optimal linear interpolation of a functional that depends on the unknown values of a vector-valued harmonizable symmetric alpha-stable random sequence from observations of the sequence with noise. We use the classical approach to derive formulas for computing values of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional. The crucial assumption of this approach is that the spectral densities of the involved stochastic sequences are exactly known. However, in practice, complete information on the spectral densities is impossible in most cases. In this situation, one finds a parametric or non-parametric estimate of the unknown spectral density and then applies one of the traditional estimation methods provided that the selected density is the true one. This procedure can result in a significant increase in the value of the estimation error. To avoid this effect, one can search for the estimates that are optimal for all densities from a certain class of admissible spectral densities. These estimates are called minimax because they minimize the maximal values of the errors of estimates for all densities from a given class. Therefore, in the case of spectral uncertainty, we use the minimax approach and propose formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of the functional for some classes of admissible spectral densities.

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