Parametric inference for discrete observations of diffusion processes with mixed effects

Let (Xt) be solution of a one-dimensional McKean-Vlasov stochastic differential equation with classical drift term V (α, x), self-stabilizing term Φ(β, x) and small noise amplitude ε. Our aim is to study the estimation of the unknown parameters α, β from a continuous observation of (Xt, t ∈ [0, T ]) under the double asymptotic framework ε tends to 0 and T tends to infinity. After centering and normalization of the process, uniform bounds for moments with respect to t ≥ 0 and ε are derived. We then build an explicit approximate log-likelihood leading to consistent and asymptotically Gaussian estimators, under the condition that ε √ T tends to 0, with original rates of convergence: the rate for the estimation of α is either ε −1 or √ T ε −1 , the rate for the estimation of β is √ T . Moreover, the estimators are asymptotically efficient.

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“…[42]), diffusions with mixed effects (see e.g. [14,15,45]), stochastic partial differential equations (see e.g. [1,9]).…”

Section: Introductionmentioning

confidence: 99%

Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models

Genon-Catalot

Larédo

2021

Electron. J. Statist.

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“…Mixed effects that enter the diffusion coefficient were investigated in Delattre et al . (). The case of continuous time observations of a univariate SDMEM with Gaussian and mixture of Gaussian mixed effects entering the drift linearly was considered in Delattre et al .…”

Section: Introductionmentioning

confidence: 97%

“…For discrete time observations, Hermite expansion of the transition density has been combined with Gaussian quadrature algorithms and Laplace's approximation (Picchini et al, 2010;Picchini and Ditlevsen, 2011). Mixed effects that enter the diffusion coefficient were investigated in Delattre et al (2015Delattre et al ( , 2018. The case of continuous time observations of a univariate SDMEM with Gaussian and mixture of Gaussian mixed effects entering the drift linearly was considered in Delattre et al (2013Delattre et al ( , 2016 and Maitra and Bhattacharya (2018a).…”

Section: Introductionmentioning

confidence: 99%

Inference for Biomedical Data by Using Diffusion Models with Covariates and Mixed Effects

Ruse

Samson

Ditlevsen

2019

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary Neurobiological data such as electroencephalography measurements pose a statistical challenge due to low spatial resolution and poor signal‐to‐noise ratio, as well as large variability from subject to subject. We propose a new modelling framework for this type of data based on stochastic processes. Stochastic differential equations with mixed effects are a popular framework for modelling biomedical data, e.g. in pharmacological studies. Whereas the inherent stochasticity of diffusion models accounts for prevalent model uncertainty or misspecification, random‐effects model intersubject variability. The two‐layer stochasticity, however, renders parameter inference challenging. Estimates are based on the discretized continuous time likelihood and we investigate finite sample and discretization bias. In applications, the comparison of, for example, treatment effects is often of interest. We discuss hypothesis testing and evaluate by simulations. Finally, we apply the framework to a statistical investigation of electroencephalography recordings from epileptic patients. We close the paper by examining asymptotics (the number of subjects going to ∞) of maximum likelihood estimators in multi‐dimensional, non‐linear and non‐homogeneous stochastic differential equations with random effects and included covariates.

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“…It is natural to think that the model parameters may vary from animal to animal, and so SDE models with fixed parameters as the one presented in (1) may not be suitable models for these applications. Parameter estimation for models where the parameters are considered random, known as mixed models or mixed-effects models, is presented in References [1,[5][6][7][8][9][10]. A review on the asymptotic inference of SDE mixed models can be found in Reference [11].…”

Section: Introductionmentioning

confidence: 99%

“…In References [8,10], for the general case where it is not possible to obtain a closed-form expression for the likelihood function, as is the case of random β, a numerical approximation based on an Hermite expansion was applied, whereas, in Reference [9], in addition to an Hermite expansion, a Gauss-Hermite quadrature was also applied, and the parameters of the SDE mixed model were estimated by the maximum likelihood method. In References [5,6], for mixed-models with linear drift term, when a closed-form expression for the likelihood function is not possible, a different approximation technique is used, based on a discretized version of the continuous-time data likelihood function.…”

Section: Introductionmentioning

confidence: 99%

Likelihood Function through the Delta Approximation in Mixed SDE Models

et al. 2022

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Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle weight. The model parameters are α, the average transformed weight at maturity, β, a growth parameter, and σ, a measure of environmental fluctuations intensity. We briefly review our previous work on estimation and prediction issues for this model and some generalizations, considering fixed parameters. In order to incorporate individual characteristics of the animals, we now consider that the parameters α and β are Gaussian random variables varying from animal to animal, which results in SDE mixed models. We estimate parameters by maximum likelihood, but, since a closed-form expression for the likelihood function is usually not possible, we approximate it using our proposed delta approximation method. Using simulated data, we estimate the model parameters and compare them with existing methodologies, showing that the proposed method is a good alternative. It also overcomes the existing methodologies requirement of having all animals weighed at the same ages; thus, we apply it to real data, where such a requirement fails.

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Parametric inference for discrete observations of diffusion processes with mixed effects

Abstract: Stochastic differential equations with mixed effects provide means to model intraindividual and interindividual variability in biomedical experiments based on longitudinal data. We consider N i.i.d. is assessed on simulated data for various models comprised in our framework.

Cited by 26 publications

References 18 publications

Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models

Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models

Inference for Biomedical Data by Using Diffusion Models with Covariates and Mixed Effects

Likelihood Function through the Delta Approximation in Mixed SDE Models

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