ROBUST ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES
MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKATheory 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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