### ROBUST ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES

**MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA**

*Theory of Stochastic Processes*

*Vol. 12 (28), no. 3–4, 2006, pp. 88–113*

We deal with the problem of optimal linear estimation of the functional $A_L\vec{\xi}=\int_0^L \vec{a}(t)\vec{\xi}(t)dt$ which depends on the unknown values of a multidimensional stationary stochastic process $\vec{\xi}(t)$ with the spectral density $F(\lambda)$ based on observations of the process $\vec{\xi}(t)+\vec{\eta}(t)$ for $t\in R\setminus [0,L]$, where $\vec{\eta}(t)$ is uncorrelated with $\vec{\xi}(t)$ multidimensional stationary process with the spectral density $G(\lambda)$ (interpolation problem), and the problem of optimal linear estimation of the functional $A\vec{\xi}=\int_0^\infty \vec{a}(t)\vec{\xi}(t)dt$ which depends on the unknown values of a multidimensional stationary stochastic process $\vec{\xi}(t)$, $t\geq 0$, from observations of the process $\vec{\xi}(t)+\vec{\eta}(t)$ for $t<0$ (extrapolation problem). Formulas are obtained for calculation the mean square errors and the spectral characteristics of the optimal estimates of the functionals under the condition that the spectral density matrix $F(\lambda)$ of the signal process $\vec{\xi}(t)$ and the spectral density matrix $G(\lambda)$ of the noise process $\vec{\eta}(t)$ are known. The least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of the functionals are found for concrete classes $D=D_F\times D_G$ of spectral densities under the condition that spectral density matrices $F(\lambda)$ and $G(\lambda)$ are not known, but classes $D=D_F\times D_G$ of admissible spectral densities are given.

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