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Approximate maximum likelihood estimation for stochastic differential equations with random effects in the drift and the diffusion

Author

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  • Maud Delattre

    (Université Paris-Saclay)

  • Valentine Genon-Catalot

    (UMR CNRS 8145, Laboratoire MAP5, Université Paris Descartes, Sorbonne Paris Cité)

  • Catherine Larédo

    (INRA, MaIAGE
    LPMA, Paris Diderot, Sorbonne Paris Cité)

Abstract

Consider N independent stochastic processes $$(X_i(t), t\in [0,T])$$ ( X i ( t ) , t ∈ [ 0 , T ] ) , $$i=1,\ldots , N$$ i = 1 , … , N , defined by a stochastic differential equation with random effects where the drift term depends linearly on a random vector $$\Phi _i$$ Φ i and the diffusion coefficient depends on another linear random effect $$\Psi _i$$ Ψ i . For these effects, we consider a joint parametric distribution. We propose and study two approximate likelihoods for estimating the parameters of this joint distribution based on discrete observations of the processes on a fixed time interval. Consistent and $$\sqrt{N}$$ N -asymptotically Gaussian estimators are obtained when both the number of individuals and the number of observations per individual tend to infinity. The estimation methods are investigated on simulated data and show good performances.

Suggested Citation

  • Maud Delattre & Valentine Genon-Catalot & Catherine Larédo, 2018. "Approximate maximum likelihood estimation for stochastic differential equations with random effects in the drift and the diffusion," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(8), pages 953-983, November.
  • Handle: RePEc:spr:metrik:v:81:y:2018:i:8:d:10.1007_s00184-018-0666-z
    DOI: 10.1007/s00184-018-0666-z
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    References listed on IDEAS

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    1. Umberto Picchini & Andrea De Gaetano & Susanne Ditlevsen, 2010. "Stochastic Differential Mixed‐Effects Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 67-90, March.
    2. Charlotte Dion, 2016. "Nonparametric estimation in a mixed-effect Ornstein–Uhlenbeck model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(8), pages 919-951, November.
    3. L. Nie, 2006. "Strong Consistency of the Maximum Likelihood Estimator in Generalized Linear and Nonlinear Mixed-Effects Models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 63(2), pages 123-143, April.
    4. Picchini, Umberto & Ditlevsen, Susanne, 2011. "Practical estimation of high dimensional stochastic differential mixed-effects models," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1426-1444, March.
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    Cited by:

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