A contrast estimator for completely or partially observed hypoelliptic diffusion

Samson, Adeline; Thieullen, Michèle

doi:10.1016/j.spa.2012.04.006

Cited by 29 publications

(51 citation statements)

References 15 publications

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“…The value

{\overset{false~}{σ}}_{0}

is biased because of the approximation of U i , as shown by Samson and Thieullen (). Therefore, we apply the bias correction that was suggested by Samson and Thieullen () and use

{\overset{σ}{true}}_{0} = \sqrt \frac{3}{2} {\overset{σ}{true}}_{0}

as initial value.…”

Section: Partial Observationsmentioning

confidence: 98%

Hypoelliptic Diffusions: Filtering and Inference from Complete and Partial Observations

Ditlevsen

Samson

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, because of the noise structure, where the noise components of the different co‐ordinates of the multi‐dimensional process operate on different timescales, standard inference tools are ill conditioned. We propose to use a higher order scheme to approximate the likelihood, such that the different timescales are appropriately accounted for. We show consistency and asymptotic normality with non‐typical convergence rates. When only partial observations are available, we embed the approximation in a filtering algorithm for the unobserved co‐ordinates and use this as a building block in a stochastic approximation expectation–maximization algorithm. We illustrate on simulated data from three models: the harmonic oscillator, the FitzHugh–Nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input.

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“…The value

{\overset{false~}{σ}}_{0}

is biased because of the approximation of U i , as shown by Samson and Thieullen (). Therefore, we apply the bias correction that was suggested by Samson and Thieullen () and use

{\overset{σ}{true}}_{0} = \sqrt \frac{3}{2} {\overset{σ}{true}}_{0}

as initial value.…”

Section: Partial Observationsmentioning

confidence: 98%

Hypoelliptic Diffusions: Filtering and Inference from Complete and Partial Observations

Ditlevsen

Samson

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…The result of Theorem 4.7 has to be compared to the one in Theorem 2 in [28]. In [28], the estimator is obtained by minimizing a contrast.…”

Section: Fluctuation-dissipation Relation and Langevin Dynamicsmentioning

confidence: 99%

“…The result of Theorem 4.7 has to be compared to the one in Theorem 2 in [28]. In [28], the estimator is obtained by minimizing a contrast. More precisely, the authors in [28] define the contrast to minimize as L n (σ 2 ) = n−2 p=1 3 2 (X (p+1)hn − 2X phn + X (p−1)hn ) 2 h 3 n σ 2 + (n − 2) log(σ 2 ), and they obtaiñ They obtain the same rate of convergence but with the asymptotic variance equal to 9 4 σ 4 .…”

Section: Fluctuation-dissipation Relation and Langevin Dynamicsmentioning

confidence: 99%

“…In [28], the estimator is obtained by minimizing a contrast. More precisely, the authors in [28] define the contrast to minimize as L n (σ 2 ) = n−2 p=1 3 2 (X (p+1)hn − 2X phn + X (p−1)hn ) 2 h 3 n σ 2 + (n − 2) log(σ 2 ), and they obtaiñ They obtain the same rate of convergence but with the asymptotic variance equal to 9 4 σ 4 . Our definition (3.1) of the double increment of X, which is different from theirs, allows to recover the asymptotic variance 2σ 4 they get for the case of complete observations.…”

Section: Fluctuation-dissipation Relation and Langevin Dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

Estimation for stochastic damping Hamiltonian systems under partial observation. III. Diffusion term

Cattiaux¹,

León²,

Prieur³

2016

Ann. Appl. Probab.

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International audienceThis paper is the third part of our study started with Cattiaux, León and Prieur (2013, 2014). For some ergodic hamiltonian systems we obtained a central limit theorem for a non-parametric estimator of the invariant density (Cattiaux et al., 2014) and of the drift term (Cattiaux et al., 2013), under partial observation (only the positions are observed). Here we obtain similarly a central limit theorem for a non-parametric estimator of the diffusion term

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“…Actually some works have already been done, in the hypo-elliptic context, but in a parametric framework. We refer to the recent work by Samson and Thieullen (2012) and the bibliography therein.…”

Section: Introductionmentioning

confidence: 99%

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

Cattiaux¹,

León²,

Prieur³

2014

Stochastic Processes and their Applications

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A contrast estimator for completely or partially observed hypoelliptic diffusion

Cited by 29 publications

References 15 publications

Hypoelliptic Diffusions: Filtering and Inference from Complete and Partial Observations

Hypoelliptic Diffusions: Filtering and Inference from Complete and Partial Observations

Estimation for stochastic damping Hamiltonian systems under partial observation. III. Diffusion term

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

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