Theory of Stochastic Processes Vol.10 (26), no.1–2, 2004, pp.141–154 MIKHAIL MOKLYACHUK AND NATALIYA SHCHESTYUK ESTIMATION PROBLEMS FOR RANDOM FIELDS FROM OBSERVATIONS IN DISCRETE MOMENTS OF TIME 1 Problems of estimation of the unknown values of a homogeneous random field ξ (s, t) ,s ∈ R, t ∈ R from observations of the field ξ (u, v)+ η (u, v) for (u, v) ∈ T × S are investigated. Formulas are proposed for calculation the mean square errors and spectral char- acteristic of the optimal linear estimate for various sets T and S . The least favourable spectral densities and the minimax-robust spec- tral characteristic of the optimal linear estimates are found for some classes of random fields. 1. Introduction Problems of estimation of the unknown values of random fields are generalization of problems of extrapolation, interpolation and filtering of stochastic processes. Such problems arise when we are finding solutions to important problems in econometrics, automatic control, etc. Methods of solution of linear estimation problems for stochastic processes and random fields were developed by A. N. Kolmogorov [1], N. Wiener [2], A. M. Yaglom [3,4], M. I. Yadrenko [5]. Traditional methods of solution of these problems are employed under the condition that spectral densities are known exactly. In practice, however, complete information on spectral densities is impos- sible in most cases. To solve the problem the parametric or nonparametric estimates of the unknown spectral densities are found or these densities are selected by other reasoning. Then the traditional estimation method is applied provided that the estimated or selected densities are the true one. This procedure can result in a significant increasing of the value of the er- ror as K. S. Vastola and H. V. Poor [6] have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities from a certain class of the admissible spectral densities. 1 Invited lecture. 2000 Mathematics Subject Classifications. 60G60, 60M40, 60G35, 93E10, 93E11. Key words and phrases. Random fields, mean square errors, spectral characteris- tic, optimal linear estimate, least favourable spectral densities, minimax-robust spectral characteristic 154