Interpolation of stationary sequences observed with a noise

Abstract

The problem of optimal linear estimation of the functionalwhich depends on unknown values of a stochastic stationary sequence ξ(j) with the help of observations of the sequence ξ(j) + η(j) at points j ∈ Z \ S, where S = s−1 l=0 {M l , . . . , M l + N l+1 }, is considered under the assumption that the sequences {ξ(j)} and {η(j)} are mutually uncorrelated. Formulas for calculating the mean-square error and spectral characteristic of the optimal linear estimator of the functional are proposed under the condition of spectral certainty, where both spectral densities of the sequences ξ(j) and η(j) are known. The minimax (robust) method of estimation is applied in the case where the spectral densities of the sequences ξ(j) and η(j) are not known exactly, but the sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.

Minimax-Robust Estimation Problems for Stationary Stochastic Sequences

“…converges in the mean square, and then, by (38), relation (37) will be established as soon as we show that…”

Minimax-robust estimation problems for sequences with periodically stationary increments observed with noise

“…Processes with stationary increments are investigated by Luz and Moklyachuk [19,20]. We also mention works by Moklyachuk and Sidei [27,28], Masyutka, Moklyachuk and Sidei [21], who derive minimax estimates of stationary processes from observations with missed values. Moklyachuk and Kozak [17] studied the problem of interpolation of stochastic sequences with periodically stationary increments.…”

Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

“…Prediction problem for stationary sequences with missing observations is investigated in papers by Bondon [1,2], Cheng, Miamee and Pourahmadi [5], Cheng and Pourahmadi [6], Kasahara, Pourahmadi and Inoue [15], Pourahmadi, Inoue and Kasahara [33], Pelagatti [32]. In papers by Moklyachuk and Sidei [28] - [31] an approach is developed to investigation of the interpolation, extrapolation and filtering problems for stationary stochastic sequences with missing observations.…”