Minimax prediction of random processes with stationary increments from observations with stationary noise

Luz, Maksym; Moklyachuk, Mikhail

doi:10.1080/23311835.2015.1133219

Cited by 4 publications

(7 citation statements)

References 18 publications

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“…The minimax spectral characteristic h 0 = h τ (f 0 , g 0 ) is evaluated by equality (14) if h τ (f 0 , g 0 ) ∈ H D .…”

Section: The Minimax (Robust) Methods Of Interpolationmentioning

confidence: 99%

“…where g 2 (λ) is established from relation (31), is the least favorable among spectral densities of the class D u v for the optimal linear interpolation of the functional A T ξ if the function f 0 (λ) + (1 + λ 2 ) −n λ 2n g(λ) determines a solution of optimization problem (23). The function h τ (f 0 , g) defined by equality (14) is the minimax (robust) spectral characteristic of the optimal estimate of the functional A T ξ. where γ 1 is such that γ 1 (λ) ≤ 0 and γ 1 (λ) = 0 if f 0 (λ) ≥ v(λ); γ 2 is such that γ 2 (λ) ≥ 0 and γ 2 (λ) = 0 if f 0 (λ) ≤ u(λ). These results lead to the following corollary.…”

Section: Then the Functionmentioning

confidence: 99%

“…is the least favorable in the class D 0 1/f for the linear interpolation of the functional A T ξ if the function f 0 (λ) + (1 + λ 2 ) −n λ 2n g(λ) determines a solution of extremum problem (23). The function h τ (f 0 , g) defined by equality (14) is the minimax spectral characteristic of the optimal estimate of the functional A T ξ.…”

Section: The Least Favorable Spectral Densities In the Classmentioning

confidence: 99%

“…Some classical and minimax problems of extrapolation, interpolation, and filtration for stochastic processes with stationary increments are considered by Luz and Moklyachuk [11]- [14].…”

Section: Introductionmentioning

confidence: 99%

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Minimax interpolation of stochastic processes with stationary increments from observations with noise

Luz

Moklyachuk

2017

Theor. Probability and Math. Statist.

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We deal with the problem of optimal estimation of the linear functionals constructed from the missed values of a continuous time stochastic process ξ(t) with periodically stationary increments at points t ∈ [0; (N + 1)T ] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions {ξj , j ∈ Z} or corresponding to it an (infinite dimensional) vector stationary increment sequence { ξIn the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

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“…The minimax spectral characteristic h 0 = h τ (f 0 , g 0 ) is evaluated by equality (14) if h τ (f 0 , g 0 ) ∈ H D .…”

Section: The Minimax (Robust) Methods Of Interpolationmentioning

confidence: 99%

Section: Then the Functionmentioning

confidence: 99%

Section: The Least Favorable Spectral Densities In the Classmentioning

confidence: 99%

“…Some classical and minimax problems of extrapolation, interpolation, and filtration for stochastic processes with stationary increments are considered by Luz and Moklyachuk [11]- [14].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Minimax interpolation of stochastic processes with stationary increments from observations with noise

Luz

Moklyachuk

2017

Theor. Probability and Math. Statist.

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“…See the book by Golichenko and Moklyachuk (2014) for more relative results and references. The minimax-robust extrapolation, interpolation and filtering problems for stochastic sequences and processes with nth stationary increments were solved by Luz and Moklyachuk (2012, 2015a, 2015b, 2015c, 2016a, 2016bMoklyachuk & Luz, 2013). The obtained results are applied to find solution of the extrapolation and filtering problems for cointegrated sequences Moklyachuk, 2014b, 2015c).…”

Section: Introductionmentioning

confidence: 99%

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

Luz

Moklyachuk

2016

Cogent Mathematics

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The problem of optimal estimation of the linear functional A = ∑ ∞ k=0 a(k) (−k) depending on the unknown values of a stochastic sequence (m) with nth stationary increments from observations of the sequence (m) + (m) at points m = 0, −1, −2, …, where (m) is a stationary sequence uncorrelated with (m), is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are derived in the case where spectral densities of stochastic sequences are exactly known and admit the canonical factorizations. In the case of spectral uncertainty, where spectral densities are not known exactly, but sets of admissible spectral densities are specified, the minimax-robust method is applied. Formulas and relations that determine the least favourable spectral densities and the minimax-robust spectral characteristics are proposed for the given sets of admissible spectral densities. The filtering problem for a class of cointegrated sequences is investigated.The crucial assumption of application of traditional methods of finding solution to the filtering problem for random processes is that spectral densities of the processes are exactly known. However, in practical situations complete information on spectral densities is impossible and the established results cannot be directly applied to practical filtering problems. This is a reason to apply the minimax-robust method of filtering and derive the minimax estimates since they minimize the maximum value of the mean-square errors for all spectral densities from a given set of admissible densities simultaneously. In this article, we deal with the problem of optimal estimation of functionals depending on the unknown values of a random process with stationary increments based on observations of the process and a noise. In the case where spectral densities of the processes are not exactly known, relations for determining least favourable spectral densities and minimax-robust spectral characteristics are proposed.

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References

2019

Estimation of Stochastic Processes With Stationary Increments and Cointegrated Sequences

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Minimax prediction of random processes with stationary increments from observations with stationary noise

Cited by 4 publications

References 18 publications

Minimax interpolation of stochastic processes with stationary increments from observations with noise

Minimax interpolation of stochastic processes with stationary increments from observations with noise

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

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