Extrapolation Problem for Stationary Sequences with Missing Observations

Abstract

In this paper, we consider the problem of the mean square optimal estimation of linear functionals which depend on unknown values of a stationary stochastic sequence based on observations of the sequence with a stationary noise. Formulas 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 spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible densities.

Extrapolation problem for periodically correlated stochastic sequences with missing observations

“…Moklyachuk, O.Yu. Masyutka and M.I.Sidei [19], [21], [28], [29]. The interpolation problem of linear functionals from periodically correlated stochastic sequences with missing observations was investigated by I.I.…”

Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments

“…Processes with stationary increments are investigated by Moklyachuk and Luz [31,32]. We also mention works by Moklyachuk and Sidei [37,38], who derive minimax estimates of stationary processes from observations with missed values. Moklyachuk and Kozak [29] studied interpolation problem for 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.…”

“…Let the minimality condition(12) hold true. The least favorable spectral densitiesF 0 (λ), G 0 (λ) in the classes D 0 × D U Vfor the optimal linear extrapolation of the functional A ⃗ ξ are determined by relations(30),(31) for the first pairD 1 0 × D U V1 of sets of admissible spectral densities; (32),(33) for the second pairD 2 0 × D U V2 of sets of admissible spectral densities; (34),(35) for the third pairD 3 0 × D U V3 of sets of admissible spectral densities; (36),…”