Game theory and convex optimization methods in robust estimation problems (Q2740086)

The article deals with the problem of the mean square optimal estimate of the functional \(A\xi=\int_{0}^{\infty}\langle a(t),\xi(t)\rangle dt\) from observations of the process \(\xi(t)+\eta(t)\) for \(t<0\), where \(\xi(t), \;t\in R^1\), is a Hilbert space valued stationary stochastic process with spectral density \(f(\lambda)\), and \(\eta(t)\) is an uncorrelated with \(\xi(t)\) Hilbert space valued stationary stochastic process with spectral density \(g(\lambda)\).NEWLINENEWLINENEWLINELet \(h(\lambda)\) be a spectral characteristic of the optimal linear estimate \(\widehat A\xi\) of the functional \(A\xi\). Formulas for the mean square error \(\Delta(h;f,g)=E|A\xi-\widehat A\xi|^2\) and the spectral characteristic \(h(\lambda)\) are presented. The minimax spectral characteristic and least favorable spectral densities are obtained for various classes of spectral densities.