Estimation problem for continuous time stochastic processes with periodically correlated increments
Abstract: We deal with the problem of optimal estimation of the linear functionals constructed from unobserved values of a continuous time stochastic process with periodically correlated increments based on past observations of this process. To solve the problem, we construct a corresponding to the process sequence of stochastic functions which forms an infinite dimensional vector stationary increment sequence. In the case of known spectral density of the stationary increment sequence, we obtain formulas for calculating… Show more
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 1 publication
References 20 publications
Minimax interpolation of continuous time stochastic processes with periodically correlated increments observed with noise
We deal with the problem of optimal estimation of linear functionals constructed from the missed values of a continuous time stochastic process ξ ( t ) {\xi(t)} with periodically stationary increments at points t ∈ [ 0 ; ( N + 1 ) T ] t\in[0;(N+1)T] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions { ξ j ( d ) ( u ) = ξ j ( d ) ( u + j T , τ ) , u ∈ [ 0 , T ) , j ∈ ℤ } {\{\xi^{(d)}_{j}(u)=\xi^{(d)}_{j}(u+jT,\tau),u\in[0,T),\,j\in\mathbb{Z}\}} is constructed. It forms an L 2 ( [ 0 , T ) ; H ) {L_{2}([0,T);H)} -valued stationary increment sequence { ξ j ( d ) , j ∈ ℤ } {\{\xi^{(d)}_{j},j\in\mathbb{Z}\}} or corresponding to it an (infinite-dimensional) vector stationary increment sequence { ξ → j ( d ) = ( ξ k j ( d ) , k = 1 , 2 , … ) ⊤ , j ∈ ℤ } {\{\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},\,j\in\mathbb{Z}\}} . In 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.
Minimax interpolation of continuous time stochastic processes with periodically correlated increments observed with noise
We deal with the problem of optimal estimation of linear functionals constructed from the missed values of a continuous time stochastic process ξ ( t ) {\xi(t)} with periodically stationary increments at points t ∈ [ 0 ; ( N + 1 ) T ] t\in[0;(N+1)T] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions { ξ j ( d ) ( u ) = ξ j ( d ) ( u + j T , τ ) , u ∈ [ 0 , T ) , j ∈ ℤ } {\{\xi^{(d)}_{j}(u)=\xi^{(d)}_{j}(u+jT,\tau),u\in[0,T),\,j\in\mathbb{Z}\}} is constructed. It forms an L 2 ( [ 0 , T ) ; H ) {L_{2}([0,T);H)} -valued stationary increment sequence { ξ j ( d ) , j ∈ ℤ } {\{\xi^{(d)}_{j},j\in\mathbb{Z}\}} or corresponding to it an (infinite-dimensional) vector stationary increment sequence { ξ → j ( d ) = ( ξ k j ( d ) , k = 1 , 2 , … ) ⊤ , j ∈ ℤ } {\{\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},\,j\in\mathbb{Z}\}} . In 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
