Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences

Masyutka

Sidei

2019

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The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic process is considered. Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the processes are not known exactly while some sets of admissible spectral densities of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral densities.

Section: Theorem 41mentioning

confidence: 99%

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

Masyutka

Sidei

2019

Self Cite

“…Corollary 2.4 Let { ⃗ ξ(j), j ∈ Z} be a multidimensional stationary stochastic sequence with the spectral density F (λ) which satisfy the minimality condition (20). Suppose that the operator B is invertible.…”

Section: Extrapolation Problem For Sequences With Missing Observationsmentioning

confidence: 99%

“…In the book by Moklyachuk and Golichenko [24] results of investigation of the interpolation, extrapolation and filtering problems for periodically correlated stochastic sequences are proposed. In their papers Luz and Moklyachuk [18] - [20] deal with the problems of estimation of functionals which depend on the unknown values of stochastic sequences with stationary increments. 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].…”

Section: Introductionmentioning

confidence: 99%

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Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

Masyutka

Sidei

2019

Self Cite

This paper focuses on the problem of the mean square optimal estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of the sequence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived under the condition of spectral certainty. 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 of the optimal estimates of functionals are proposed for some special sets of admissible densities.

“…Recent results of minimax extrapolation problems for stationary vector-valued processes and periodically correlated processes belong to Moklyachuk and Masyutka [35,36] and Moklyachuk and Golichenko (Dubovetska) [7] respectively. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Luz

2020

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We introduce a stochastic sequence $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of the stochastic sequence $\zeta(k)$ based on its observations at points $ k<0$. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.