Minimax Prediction Problem for Multidimensional Stationary Stochastic Processes

2015

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The problem of optimal estimation of the linear functionals Aξ = ∑ ∞ k=0 a(k)ξ (k) andwhich depend on the unknown values of a stochastic sequence ξ(m) with stationary nth increments is considered. Estimates are obtained which are based on observations of the sequence ξ(m) + η(m) at points of time m = −1, −2, . . ., where the sequence η(m) is stationary and uncorrelated with the sequence ξ(m). Formulas for calculating the mean-square errors and the spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where spectral densities of the sequences ξ(m) and η(m) are exactly known. These results are applied for solving extrapolation problem for cointegrated sequences. In the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities.

Section: Lemmamentioning

confidence: 99%

Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

2015

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“…The functions f 0 (λ) and g 0 (λ) determined by equations (29), (30) are the least favorable in the class D = D f × D g for the optimal linear filtering of the functional Aξ if they determine a solution to optimization problem (25). The function h(f 0 , g 0 ) determined by formula (6) is the minimax-robust spectral characteristic of the optimal estimate of the functional Aξ.…”

Section: Theoremmentioning

confidence: 99%

“…The corresponding problems for vector-valued stationary sequences and processes are investigated by Moklyachuk and Masyutka [29], [30]. In the article by Dubovets'ka and Moklyachuk [4], [5] and in the book by Golichenko and Moklyachuk [10] the minimax estimation problems are investigated for another generalization of stationary processes -periodically correlated stochastic sequences and stochastic processes.…”

Section: Introductionmentioning

confidence: 99%

Filtering Problem for Functionals of Stationary Sequences

2016

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In this paper, we consider the problem of the mean-square optimal linear estimation of functionals which depend on the unknown values of a stationary stochastic sequence from observations with noise. In the case of spectral certainty in which the spectral densities of the sequences are exactly known, we propose formulas for calculating the spectral characteristic and value of the mean-square error of the estimate by using the Fourier coefficients of some functions from the spectral densities. When the spectral densities are not exactly known but a class of admissible spectral densities is given, the minimax-robust method of estimation is applied. Formulas for determining the least favourable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functionals are proposed for some specific classes of admissible spectral densities.Keywords Stationary stochastic sequence; optimal linear mean-square estimate; mean square error; least favourable spectral density; minimax-robust estimate; minimax-robust spectral characteristic

“…Let a spectral density f 0 (λ) ∈ D 2ε be such that the function determined by formula (26) is bounded. The condition f 0 ∈ ∂∆ D (f 0 ) gives us the equation…”

Section: Least Favourable Spectral Density In the Class D U Vmentioning

confidence: 99%

“…Later Franke [8], Franke and Poor [9] investigated the minimaxrobust extrapolation and interpolation problems for stationary sequences by using methods of convex optimization. In papers by Moklyachuk [19] - [26] the minimax approach was applied to extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes. Methods of solution the minimax-robust estimation problems for vector-valued stationary processes were developed by Moklyachuk and Masyutka [23] - [28].…”

Section: Introductionmentioning

confidence: 99%

Minimax Interpolation Problem for Random Processes with Stationary Increments

2015

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The problem of mean-square optimal estimation of the linear functional A T ξ = ∫ T 0 a(t)ξ(t)dt that depends on the unknown values of a continuous time random process ξ(t), t ∈ R, with stationary nth increments from observations of the process ξ(t) at time points t ∈ R \ [0; T ] is investigated under the condition of spectral certainty as well as under the condition of spectral uncertainty. Formulas for calculation the value of the mean-square error and spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty where spectral density of the process is exactly known. In the case of spectral uncertainty where spectral density of the process is not exactly known, but a class of admissible spectral densities is given, relations that determine the least favourable spectral density and the minimax spectral characteristic are specified.Keywords Random process with stationary increments, minimax-robust estimate, mean-square error, least favourable spectral density, minimax spectral characteristic AMS 2010 subject classifications Primary: 60G10, 60G25, 60G35, Secondary: 62M20, 93E10, 93E11