Random Oper, and Stoch. Equ.f Vol. 3, No. 1, pp. 63-74 (1995) © VSP 1995 On interpolation problem for vector-valued stochastic sequences M. P. MOKLYACHUK Department of M echanics and M athematics, Kiev University, Kiev, Ukraine Received for ROSE 15 December 1994 A bstract— The problem of the linear mean square optimal estimation is considered for the functional 3=0 of a Hilbert space valued stationary stochastic sequence £()), j £ Z from observations of the sequence £ (J) + 17 (f) for j £ Z \ {0,1 .... , .V ]. Formulas are derived for computing the mean square error and the spectral characteristic of the optimal estimate of the functional /4 V £. The minimax spectral characteristic and the least favorable spect ral densities are found for various classes of spectral densities. 1. IN T R O D U C T IO N The classical Kolmogorov method of linear interpolation, extrapolation and filtering of weakly stationary stochastic sequences [ 1], [2 ] may be employed under the condition that spectral densities of stochastic sequences are known. In practice, however, the problem of estimation of unknown values of stochastic sequence arises where the spectral density is not known exactly. To solve the problem, the parametric or nonparametric estimate of the unknown spectral density is found. Then the classical method is applied provided that the estimate of the density is the true one. This procedure can result in a significant increasing of the value of the error as Vastola and Poor have demonstrated with the help of some examples [3]. For this reason it is necessary to search the estimate of the unknown value of the stochastic sequence that has the least value of the error for all densities from a certain class of spectral densities. Such an approach to the problem of interpolation, extrapolation and filtering of stationary stochastic sequences have been taken into consideration by many investigators [3] - [23]. A survey of results in minimax (robust) methods of data processing can be found in the paper [9]. The paper [ 10 ] is the first one where the minimax interpolation problem for the e - pollution model is investigated. The relation of the minimax interpolation problem with the problem of robust hypothesis testing is indicated in [8 ]. In papers [4] - [23] the minimax interpolation, extrapolation and filtering problems are investigated with the help of the convex optimization methods. Translated by Yu. V. Kozachenko