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Parametric inference for mixed models defined by stochastic differential equations

Published online by Cambridge University Press:  23 January 2008

Sophie Donnet  and
Adeline Samson
Sophie Donnet
Affiliation:
Paris-Sud University, Laboratoire de Mathématiques, Orsay, France; UMR CNRS 8145, University Paris 5, Paris, France; adeline.samson@univ-paris5.fr
Adeline Samson
Affiliation:
INSERM U738, Paris, France; University Paris 7, Paris, France; UMR CNRS 8145, University Paris 5, Paris, France
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Abstract

Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants. A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.

Keywords

Brownian bridgediffusion processEuler-Maruyama approximationGibbs algorithmincomplete data modelmaximum likelihood estimationnon-linear mixed effects modelSAEM algorithm
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 196 - 218
Copyright
© EDP Sciences, SMAI, 2008

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References

Aït-Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70 (2002) 223–262. CrossRef
Andrieu, C. and Moulines, E., On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Prob. 16 (2006) 1462–1505. CrossRef
V. Bally and D. Talay, The law of the Euler Scheme for Stochastic Differential Equations: I. Convergence Rate of the Density. Technical Report 2675, INRIA (1995).
Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations (II): convergence rate of the density. Monte Carlo Methods Appl. 2 (1996) 93–128. CrossRef
Beal, S.L. and Sheiner, L.B., Estimating population kinetics. Crit. Rev. Biomed. Eng. 8 (1982) 195–222.
J.E. Bennet, A. Racine-Poon and J.C. Wakefield, MCMC for nonlinear hierarchical models. Chapman & Hall, London (1996) 339–358.
Beskos, A., Papaspiliopoulos, O., Roberts, G.O. and Fearnhead, P., Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. B 68 (2006) 333–382. CrossRef
Beskos, A. and Roberts, G.O., Exact simulation of diffusions. Ann. Appl. Prob. 15 (2005) 2422–2444. CrossRef
Bibby, B.M. and Sørensen, M., Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 (1995) 17–39. CrossRef
Celeux, G. and Diebolt, J., The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Computational. Statistics Quaterly 2 (1985) 73–82.
D. Dacunha-Castelle and M. Duflo, Probabilités et statistiques. Tome 2. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree]. Masson, Paris, 1983. Problèmes à temps mobile. [Movable-time problems].
Dacunha-Castelle, D. and Florens-Zmirou, D., Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19 (1986) 263–284. CrossRef
Delyon, B., Lavielle, M. and Moulines, E., Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94–128.
Dembo, A. and Zeitouni, O., Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm. Stoch. Process. Appl. 23 (1986) 91–113. CrossRef
Dempster, A P., Laird, N.M. and Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 (1977) 1–38. With discussion.
Ditlevsen, S. and De Gaetano, A., Mixed effects in stochastic differential equation models. REVSTAT- Statistical Journal 3 (2005) 137–153.
S. Donnet and A. Samson, Estimation of parameters in incomplete data models defined by dynamical systems. J. Stat. Plan. Inf. (2007).
Douc, R. and Matias, C., Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381–420. CrossRef
Elerian, O., Chib, S. and Shephard, N., Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 (2001) 959–993. CrossRef
Eraker, B., MCMC analysis of diffusion models with application to finance. J. Bus. Econ. Statist. 19 (2001) 177–191. CrossRef
Genon-Catalot, V. and Jacod, J., On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119–151.
Gloter, A. and Jacod, J., Diffusions with measurement errors. I. Local asymptotic normality. ESAIM: PS 5 (2001) 225–242. CrossRef
Gloter, A. and Jacod, J., Diffusions with measurement errors. II. Optimal estimators. ESAIM: PS 5 (2001) 243–260 (electronic). CrossRef
Kessler, M., Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 (1997) 211–229. CrossRef
R. Krishna, Applications of Pharmacokinetic principles in drug development. Kluwer Academic/Plenum Publishers, New York (2004).
Kuhn, E. and Lavielle, M., Coupling a stochastic approximation version of EM with a MCMC procedure. ESAIM: PS 8 (2004) 115–131. CrossRef
Kuhn, E. and Lavielle, M., Maximum likelihood estimation in nonlinear mixed effects models. Comput. Statist. Data Anal. 49 (2005) 1020–1038. CrossRef
S. Kusuoka and D. Stroock, Applications of the malliavin calculus, part II. J. Fac. Sci. Univ. Tokyo. Sect. IA, Math. 32 (1985) 1–76.
T. Kutoyants, Parameter estimation for stochastic processes. Helderman Verlag Berlin (1984).
Lindstrom, M.L. and Bates, D.M., Nonlinear mixed effects models for repeated measures data. Biometrics 46 (1990) 673–687. CrossRef
Louis, T.A., Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 (1982) 226–233.
Overgaard, R.V., Jonsson, N., Tornøe, C.W. and Madsen, H., Non-linear mixed-effects models with stochastic differential equations: Implementation of an estimation algorithm. J Pharmacokinet. Pharmacodyn. 32 (2005) 85–107. CrossRef
Pedersen, A.R., A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22 (1995) 55–71.
Pinheiro, J.C. and Bates, D.M., Approximations to the log-likelihood function in the non-linear mixed-effect models. J. Comput. Graph. Statist. 4 (1995) 12–35.
R. Poulsen, Approximate maximum likelihood estimation of discretely observed diffusion process. Center for Analytical Finance, Working paper 29 (1999).
B.L.S. Prakasa Rao, Statistical Inference for Diffusion Type Processes. Arnold Publisher (1999).
Roberts, G.O. and Stramer, O., On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88 (2001) 603–621. CrossRef
Schweppe, F., Evaluation of likelihood function for gaussian signals. IEEE Trans. Inf. Theory 11 (1965) 61–70. CrossRef
Singer, H., Continuous-time dynamical systems with sampled data, error of measurement and unobserved components. J. Time Series Anal. 14 (1993) 527–545. CrossRef
Sørensen, H., Parametric inference for diffusion processes observed at discrete points in time: a survey. Int. Stat. Rev 72 (2004) 337–354. CrossRef
Sørensen, M., Prediction-based estimating functions. Econom. J. 3 (2000) 123–147. CrossRef
Tierney, L., Markov chains for exploring posterior distributions. Ann. Statist. 22 (1994) 1701–1762. CrossRef
Tornøe, C.W., Overgaard, R.V., Agersø, H., Nielsen, H.A., Madsen, H. and Jonsson, E.N., Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations. Pharm. Res. 22 (2005) 1247–1258. CrossRef
Wei, G.C.G. and Tanner, M.A., Calculating the content and boundary of the highest posterior density region via data augmentation. Biometrika 77 (1990) 649–652. CrossRef
Wolfinger, R., Laplace's approximation for nonlinear mixed models. Biometrika 80 (1993) 791–795. CrossRef