Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes
The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–U...
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Published in | Journal of the Korean Statistical Society Vol. 45; no. 3; pp. 329 - 341 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Singapore
Elsevier B.V
01.09.2016
Springer Singapore Elsevier 한국통계학회 |
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Abstract | The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt=θXtdt+dGt,t≥0 with an unknown parameter θ>0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimator θ˜t of θ based on the observation {Xs,s∈[0,t]} as t→∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H∈(0,1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion. |
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AbstractList | The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt = θXt dt + dGt , t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimatorθt of θ based on the observation {Xs, s ∈ [0, t]} as t → ∞.
Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion. KCI Citation Count: 1 The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck process defined as dX t = θX t dt + dG t , t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G , ensuring the strong consistency and the asymptotic distribution of our estimator θ̃ t of θ based on the observation { X s , s ∈ [0, t ]} as t → ∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion. The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt=θXtdt+dGt,t≥0 with an unknown parameter θ>0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimator θ˜t of θ based on the observation {Xs,s∈[0,t]} as t→∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H∈(0,1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion. |
Author | El Machkouri, Mohamed Es-Sebaiy, Khalifa Ouknine, Youssef |
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Keywords | secondary Parameter estimation Non-ergodic Gaussian Ornstein–Uhlenbeck process primary Non-ergodic Gaussian Ornstein-Uhlenbeck process secondary 60G18 primary 62F12 |
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article-title: Parametric estimation for sub-fractional Ornstein–Uhlenbeck process publication-title: Journal of Statistical Planning and Inference contributor: fullname: Mendy – volume: 5 start-page: 333 year: 2011 end-page: 340 ident: br000015 article-title: An extension of bifractional Brownian motion publication-title: Communications on Stochastic Analysis contributor: fullname: Es-Sebaiy – volume: 35 start-page: 1183 year: 2007 end-page: 1212 ident: br000080 article-title: Statistical aspects of the fractional stochastic calculus publication-title: The Annals of Statistics contributor: fullname: Viens – volume: 1 start-page: 1 year: 2011 end-page: 16 ident: br000020 article-title: Parameter estimation for fractional Ornstein–Uhlenbeck processes: Non-ergodic case publication-title: Frontiers in Science and Engineering contributor: fullname: Ouknine – volume: 80 start-page: 1030 year: 2010 end-page: 1038 ident: br000050 article-title: Parameter estimation for fractional Ornstein–Uhlenbeck processes publication-title: Statistics & Probability Letters contributor: fullname: Nualart – year: 2013 ident: br000005 article-title: Drift parameter estimation for fractional Ornstein–Uhlenbeck process of the second kind publication-title: Statistics: A Journal of Theoretical and Applied Statistics contributor: fullname: Morlanes – volume: Vol. 34 start-page: 385 year: 2013 end-page: 412 ident: br000040 article-title: Parameter estimation for publication-title: Springer proceedings in mathematics and statistics contributor: fullname: Nourdin – start-page: 251 year: 1936 end-page: 282 ident: br000085 article-title: An inequality of the Hölder type connected with Stieltjes integration publication-title: Acta Mathematica contributor: fullname: Young – volume: Vol. 4 year: 2012 ident: br000075 article-title: Selected aspects of fractional Brownian motion publication-title: Bocconi & springer series contributor: fullname: Nourdin – volume: 5 start-page: 229 year: 2002 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ident: CR2 article-title: Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind publication-title: Statistical Inference for Stochastic Processes doi: 10.1007/s11203-014-9111-8 contributor: fullname: Viitasaari – volume: 35 start-page: 285 issue: 2 year: 2015 end-page: 300 ident: CR5 article-title: Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein-Uhlenbeck processes publication-title: Probability and Mathematical Statistics contributor: fullname: Es-Sebaiy – year: 2015 ident: CR9 publication-title: Parameter estimation for SDEs related to stationary Gaussian processes, preprint contributor: fullname: Viens – volume: 1 start-page: 1 issue: 1 year: 2011 end-page: 16 ident: CR4 article-title: Parameter estimation for fractional Ornstein-Uhlenbeck processes: Non-ergodic case publication-title: Frontiers in Science and Engineering contributor: fullname: Ouknine – start-page: 427 year: 2013 end-page: 442 ident: CR11 article-title: Parameter estimation for fractional Ornstein-Uhlenbeck processes with discrete observations publication-title: Springer proceedings in mathematics and statistics: Vol. 34. 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start-page: 1183 issue: 3 year: 2007 end-page: 1212 ident: CR16 article-title: Statistical aspects of the fractional stochastic calculus publication-title: The Annals of Statistics doi: 10.1214/009053606000001541 contributor: fullname: Viens – year: 2015 ident: CR13 publication-title: Bifractional Brownian motion: Existence and Border cases, preprint contributor: fullname: Volkova – start-page: 385 year: 2013 end-page: 412 ident: CR8 article-title: Parameter estimation for α-fractional bridges publication-title: Springer proceedings in mathematics and statistics: Vol. 34 contributor: fullname: Nourdin – volume: 5 start-page: 333 year: 2011 end-page: 340 ident: CR3 article-title: An extension of bifractional Brownian motion publication-title: Communications on Stochastic Analysis doi: 10.31390/cosa.5.2.05 contributor: fullname: Es-Sebaiy – start-page: 251 year: 1936 end-page: 282 ident: CR17 article-title: An inequality of the Hölder type connected with Stieltjes integration publication-title: Acta Mathematica contributor: fullname: Young – volume: 5 start-page: 229 year: 2002 ident: 10.1016/j.jkss.2015.12.001_br000060 article-title: Statistical analysis of the fractional Ornstein–Uhlenbeck type process publication-title: Statistical Inference for Stochastic Processes doi: 10.1023/A:1021220818545 contributor: fullname: Kleptsyna – volume: Vol. 4 year: 2012 ident: 10.1016/j.jkss.2015.12.001_br000075 article-title: Selected aspects of fractional Brownian motion contributor: fullname: Nourdin – ident: 10.1016/j.jkss.2015.12.001_br000045 – volume: 80 start-page: 1030 year: 2010 ident: 10.1016/j.jkss.2015.12.001_br000050 article-title: Parameter estimation for fractional Ornstein–Uhlenbeck processes publication-title: Statistics & Probability Letters doi: 10.1016/j.spl.2010.02.018 contributor: fullname: Hu – volume: 35 start-page: 285 issue: 2 year: 2015 ident: 10.1016/j.jkss.2015.12.001_br000025 article-title: Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein–Uhlenbeck processes publication-title: Probability and Mathematical Statistics contributor: fullname: Cénac – volume: 1 start-page: 1 issue: 1 year: 2011 ident: 10.1016/j.jkss.2015.12.001_br000020 article-title: Parameter estimation for fractional Ornstein–Uhlenbeck processes: Non-ergodic case publication-title: Frontiers in Science and Engineering contributor: fullname: Belfadli – volume: Vol. 34 start-page: 385 year: 2013 ident: 10.1016/j.jkss.2015.12.001_br000040 article-title: Parameter estimation for α-fractional bridges contributor: fullname: Es-Sebaiy – volume: 18 start-page: 205 issue: 3 year: 2015 ident: 10.1016/j.jkss.2015.12.001_br000010 article-title: Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind publication-title: Statistical Inference for Stochastic Processes doi: 10.1007/s11203-014-9111-8 contributor: fullname: Azmoodeh – year: 2013 ident: 10.1016/j.jkss.2015.12.001_br000005 article-title: Drift parameter estimation for fractional Ornstein–Uhlenbeck process of the second kind publication-title: Statistics: A Journal of Theoretical and Applied Statistics contributor: fullname: Azmoodeh – volume: 5 start-page: 333 year: 2011 ident: 10.1016/j.jkss.2015.12.001_br000015 article-title: An extension of bifractional Brownian motion publication-title: Communications on Stochastic Analysis doi: 10.31390/cosa.5.2.05 contributor: fullname: Bardina – ident: 10.1016/j.jkss.2015.12.001_br000035 – ident: 10.1016/j.jkss.2015.12.001_br000065 – volume: 35 start-page: 1183 issue: 3 year: 2007 ident: 10.1016/j.jkss.2015.12.001_br000080 article-title: Statistical aspects of the fractional stochastic calculus publication-title: The Annals of Statistics doi: 10.1214/009053606000001541 contributor: fullname: Tudor – start-page: 251 year: 1936 ident: 10.1016/j.jkss.2015.12.001_br000085 article-title: An inequality of the Hölder type connected with Stieltjes integration publication-title: Acta Mathematica doi: 10.1007/BF02401743 contributor: fullname: Young – volume: Vol. 34 start-page: 427 year: 2013 ident: 10.1016/j.jkss.2015.12.001_br000055 article-title: Parameter estimation for fractional Ornstein–Uhlenbeck processes with discrete observations contributor: fullname: Hu – ident: 10.1016/j.jkss.2015.12.001_br000030 – volume: 143 start-page: 663 year: 2013 ident: 10.1016/j.jkss.2015.12.001_br000070 article-title: Parametric estimation for sub-fractional Ornstein–Uhlenbeck process publication-title: Journal of Statistical Planning and Inference doi: 10.1016/j.jspi.2012.10.013 contributor: fullname: Mendy |
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SubjectTerms | Applied Statistics Bayesian Inference Mathematics Non-ergodic Gaussian Ornstein–Uhlenbeck process Parameter estimation Probability Statistical Theory and Methods Statistics Statistics and Computing/Statistics Programs 통계학 |
Title | Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes |
URI | https://dx.doi.org/10.1016/j.jkss.2015.12.001 https://link.springer.com/article/10.1016/j.jkss.2015.12.001 https://hal.science/hal-02375093 https://www.kci.go.kr/kciportal/ci/sereArticleSearch/ciSereArtiView.kci?sereArticleSearchBean.artiId=ART002151107 |
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ispartofPNX | Journal of the Korean Statistical Society, 2016, 45(3), , pp.329-341 |
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