Robust Extrapolation Problem for Stochastic Sequences with Stationary Increments

2015

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This survey provides an overview of optimal estimation of linear functionals which depend on the unknown values of a stationary stochastic sequence. Based on observations of the sequence without noise as well as observations of the sequence with a stationary noise, estimates could be obtained. Formulas for calculating the spectral characteristics and the mean-square errors of the optimal estimates of functionals are derived in the case of spectral certainty, where spectral densities of the sequences are exactly known. In the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics of estimates are presented for some special classes of admissible spectral densities.

Section: Minimax-robust Estimation Problems For Stationary Stochasticmentioning

confidence: 99%

Section: Conclusion Remarksmentioning

confidence: 99%

Minimax-Robust Estimation Problems for Stationary Stochastic Sequences

2015

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“…We will exploit the representation of the functional Aζ which is proposed in [25] and is described in the following lemma.…”

Section: Extrapolation Problemmentioning

confidence: 99%

“…In papers by Luz and Moklyachuk [22,25] the problem of optimal linear extrapolation of linear functionals which depend on the unknown values of stochastic sequences and random processes with nth stationary increments from the observations without noise is investigate. The classical extrapolation problem for a non-stationary sequence which is observed with a non-stationary noise was studied by Bell [1].…”

Section: Introductionmentioning

confidence: 99%

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

Luz

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.

“…Analogous minimax estimation problems (extrapolation, interpolation and filtering) for linear functionals which depend on unknown values of periodically correlated stochastic processes were investigated by Dubovets'ka and Moklyachuk [2] - [6]. Minimax-robust extrapolation, interpolation and filtering problems for random processes and sequences with stationary increments are investigated by Luz and Moklyachuk [13] - [18]. In particular, solutions to the minimax-robust interpolation problem for stochastic sequences with stationary increments are proposed in papers [13], [14].…”

Section: Introductionmentioning

confidence: 99%

Minimax Interpolation Problem for Random Processes with Stationary Increments

Luz

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