Theory of Probability and Mathematical Statistics
(Teoriya imovirnostey ta matematychna statystyka)
Minimax interpolation of stochastic processes with stationary increments from observations with noise
M. M. Luz, M. P. Moklyachuk
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Abstract: The problem of optimal estimation of the linear functional $A_T{\xi}=\int_0^Ta(t)\xi(t)dt$ depending on the unknown values of a random process $\xi(t)$ with stationary increments from observations of the process $\xi(t)+\eta(t)$ at points $t\in\mr R\setminus[0;T]$, where $\eta(t)$ is a stationary random process uncorrelated with $\xi(t)$, is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral densities are exactly known. Relations that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for the given sets of admissible spectral densities.
Keywords: Stochastic processes with stationary increments; minimax-robust estimate; mean square error; least favorable spectral density; minimax-robust spectral characteristic
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