Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences
Abstract
The problem of optimal estimation of linear functionals $A {\xi}=$ \quad $\sum_{k=0}^{\infty}a (k)\xi(-k)$ and $A_N{\xi}=\sum_{k=0}^{N}a (k)\xi(-k)$ which depend on the unknown values of a stochastic sequence $\xi(m)$ with stationary $n$th increments from observations of the sequence $\xi(m)+\eta(m)$ at points of time $m=0,-1,-2,\ldots$ is considered in the case where the sequence $\eta(m)$ is stationary and uncorrelated with the sequence $\xi(m)$.Formulas for calculating the mean-square errors and the spectral characteristics of optimal estimates of the functionals are proposed under condition of spectral certainty, where spectral densities of the sequences $\xi(k)$ and $\eta(k)$ are exactly known. The filtering problem for one class of cointegrated sequences is solved. Minimax (robust) method of estimation is applied in the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some definite sets of admissible densities.References
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