On the problem of minimax extrapolation of vector sequences perturbed by white noise. (English. Russian original) Zbl 0834.62095

Moklyachuk, M. P.

Theory Probab. Math. Stat. 46, 89-102 (1993); translation from Teor. Jmovirn. Mat. Stat. 46, 88-104 (1992).

Summary: The problem of optimal linear estimation of the transformation \[ A\xi= \sum_{j=0}^\infty \langle a(j), \xi(j) \rangle \] of a linear sequence \(\xi (j)\) is studied, based on observations of the sequence \(\xi (j)+ \eta (j)\) when \(j<0\). The stationary sequences \(\xi (j)\) and \(\eta (j)\) assume values in Hilbert space, are not correlated, and have spectral densities \(f(\lambda)\) and \(g(\lambda) =g\). The minimax spectral characteristics of the optimal estimator of the transformation \(A\xi\) and the least favorable spectral densities \(f^0 (\lambda)\in {\mathcal D}\) for particular classes of densities \({\mathcal D}\) are found.

MSC:

62M20	Inference from stochastic processes and prediction
62M15	Inference from stochastic processes and spectral analysis

Keywords:

operator-valued correlation function; moving average; optimal linear estimation; transformation; stationary sequences; Hilbert space; spectral densities; minimax spectral characteristics; least favorable spectral densities