Stochastic autoregressive sequences and minimax interpolation. (English. Ukrainian original) Zbl 0840.60032

Moklyachuk, M. P.

Theory Probab. Math. Stat. 48, 95-103 (1994); translation from Teor. Jmovirn. Mat. Stat. 48, 135-146 (1993).

Summary: The problem of the linear mean square optimal estimation is considered for the transformation \(A_N \xi = \sum^N_{j = 0} a(j) \xi (j)\) of a stationary stochastic sequence \(\xi (j)\) with density \(f(\lambda)\) from observations of the sequence \(\xi (j)\) for \(j \in \mathbb{Z} \backslash \{0,1, \dots, N\}\). The least favorable spectral densities \(f_0 (\lambda) \in {\mathcal D}\) and the minimax (robust) spectral characteristics of the optimal estimate of the transformation \(A_N \xi\) are found for various classes \({\mathcal D}\) of spectral densities. It is shown that spectral densities of the autoregressive sequences are the least favorable for the optimal estimation of the transformation \(A_N \xi\) in certain classes \({\mathcal D}\) of spectral densities.

MSC:

60G10	Stationary stochastic processes
60G25	Prediction theory (aspects of stochastic processes)
62M20	Inference from stochastic processes and prediction
93E10	Estimation and detection in stochastic control theory
62C20	Minimax procedures in statistical decision theory

Keywords:

linear mean square optimal estimation; stationary stochastic sequence; spectral characteristics; autoregressive sequences