Duals of random vectors and processes with applications to prediction problems with missing values

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.

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“…The function h(f, g 0 ) calculated by (13) is the minimax spectral characteristic of the optimal linear estimate of the functional Aξ.…”

Section: Corollary 44mentioning

confidence: 99%

“…Then the functions f 0 (λ), g 0 (λ) determined by (37), (38) are the least favorable densities in the class D u v × D ε if they determine a solution to the optimization problem (27). The function h(f 0 , g 0 ) calculated by (13) is the minimax spectral characteristic of the optimal linear estimate of the functional Aξ.…”

Section: Theorem 51mentioning

confidence: 99%

Extrapolation Problem for Stationary Sequences with Missing Observations

Moklyachuk

Sidei

2017

Stat., optim. inf. comput.

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“…is defined by formula (13). The spectral characteristic φ p (λ) of the optimal estimate ξ(p) is calculated by formula (14). The value of the mean-square error ∆(f, g; ξ(p)) is calculated by formula (15).…”

Section: Corollarymentioning

confidence: 99%

“…It follows from relations (14) and (16) that the spectral characteristic φ(λ) of the estimate ξ(0) of the element ξ(0) is of the form…”

Section: Denote By R(k)mentioning

confidence: 99%

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Filtering Problem for Functionals of Stationary Sequences

Moklyachuk

Luz

2016

Stat., optim. inf. comput.

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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

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Optimal linear interpolation of multiple missing values

McElroy

Politis

2022

Stat Inference Stoch Process

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Duals of random vectors and processes with applications to prediction problems with missing values

Cited by 22 publications

References 27 publications

Extrapolation Problem for Stationary Sequences with Missing Observations

Extrapolation Problem for Stationary Sequences with Missing Observations

Filtering Problem for Functionals of Stationary Sequences

Optimal linear interpolation of multiple missing values

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