Minimax-robust filtering problem for stochastic sequences with stationary increments

Abstract

The problem of optimal estimation is considered for the linear functional \[ A {\xi }=\sum _{k=0}^{\infty }a (k)\xi (-k),\] which depends on unknown values of a stochastic sequence $\xi (k)$ with stationary $n$-th order increments from observations of the sequence $\xi (k)+\eta (k)$ for $k=0,-1,-2,\dots$ . Formulas suitable for calculating the mean-square error and spectral characteristic of the optimal linear estimate of the above functional are derived under the condition of the spectral definiteness, that is in the case where the spectral densities of the sequences $\xi (k)$ and $\eta (k)$ are exactly known. The minimax (robust) method of estimation is applied 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 special sets of admissible spectral densities.