Robust interpolation problem for stochastic sequences with stationary increments

Abstract

The problem of optimal estimation of the functional ANξ =PN k=0a(k)ξ(k) depending on the unknown values of stochastic sequence ξ(k) with stationary nth increments from observations of the sequence ξ(k) + η(k) at points of time k = N + 1,N + 2,... and observations of the sequence ξ(k) at points of time k = −1,−2,... is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under condition of spectral certainty, where spectral densities of the sequences ξ(k) and η(k) are exactly known. Minimax (robust) method of estimation is used in the case where spectral densities are not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some definite sets of admissible densities.