Interpolation Problem for Stationary Sequences with Missing Observations

Abstract

The problem of the mean-square optimal estimation of the linear functional AsξFormulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty, where the spectral density of the sequence ξ(j) is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density is not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

Minimax-Robust Estimation Problems for Stationary Stochastic Sequences

“…For the relative results on the mean-square optimal linear interpolation of linear functionals for stationary stochastic sequences and processes see papers by Moklyachuk [50] - [57], book by Moklyachuk and Masyutka [64], papers by Moklyachuk and Sidei [67], Moklyachuk and Ostapenko [65].…”

Minimax-Robust Estimation Problems for Stationary Stochastic Sequences

“…Note, that ∥Ad∥ 2 < ∞ under the conditions (67). The spectral characteristic h(f ) of the optimal estimate is calculated by the formula…”

Extrapolation Problem for Stationary Sequences with Missing Observations

“…Let the functions defined by (31), (32) be bounded. Then the functions f 0 (λ), g 0 (λ) determined by equations (33), (34) are the least favorable spectral densities in the class D W × D 0 if they determine a solution to optimization problem (27).…”