Abstract
The penalized simulated maximum likelihood (PSML) approach can be used to estimate parameters for a stochastic differential equation model based on completely or partially observed discrete-time observations. The PSML uses an auxiliary variable importance sampler and parameters are estimated in a penalized maximum likelihood framework. In this paper, we extend the PSML to allow for measurement error, including unknown initial conditions. Simulation studies for two stochastic models and a real world example aimed at understanding the dynamics of chronic wasting disease illustrate that our method has favorable performance in the presence of measurement error. PSML reduces both the bias and root mean squared error as compared to existing methods. Lastly, we establish consistency and asymptotic normality for the proposed estimators.
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References
Aït-Sahalia Y (2002) Transition densities for interest rate and other nonlinear diffusions. J Finance 54(4):1361–1395
Aït-Sahalia Y (2008) Closed-form likelihood expansions for multivariate diffusions. Ann Stat 2(36):906–937
Bengtsson T, Snyder C, Nychka D (2003) Toward a nonlinear ensemble filter for high-dimensional systems. J Geophys Res 108(D24):8775. https://doi.org/10.1029/2002JD002900
Billingsley P (1961) Statistical inference for Markov processes, vol 2. University of Chicago Press, Chicago
Donnet S, Foulley J-L, Samson A (2010) Bayesian analysis of growth curves using mixed models defined by stochastic differential equations. Biometrics 66(3):733–741
Durham G, Gallant A (2002) Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J Bus Econ Stat 20(3):297–338
Eraker B (2001) MCMC analysis of diffusion models with application to finance. J Bus Econ Stat 19(2):177–191
Fuchs C (2013) Inference for diffusion processes: with applications in life sciences. Springer, New York
Geremia C, Miller MW, Hoeting JA, Antolin MF, Hobbs NT (2015) Bayesian modeling of prion disease dynamics in mule deer using population monitoring and capture-recapture data. PloS One 10(10):e0140687
Golightly A, Wilkinson DJ (2005) Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics 61(3):781–788
Golightly A, Wilkinson DJ (2006) Bayesian sequential inference for nonlinear multivariate diffusions. Stat Comput 16(4):323–338
Golightly A, Wilkinson DJ (2008) Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput Stat Data Anal 52(3):1674–1693
Golightly A, Wilkinson DJ (2011) Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1(6):807–820
Kloeden P, Platen E (1992) Numerical solution of stochastic differential equations. Springer, New York
Lindström E (2012) A regularized bridge sampler for sparsely sampled diffusions. Stat Comput 22(2):615–623
Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141
Miller M, Hobbs N, Tavener S (2006) Dynamics of prion disease transmission in mule deer. Ecol Appl 16(6):2208–2214
Pedersen A (1995a) Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli 1(3):257–279
Pedersen A (1995b) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand J Stat 22(1):55–71
Pitt MK, Shephard N (1999) Filtering via simulation: auxiliary particle filters. J Am Stat Assoc 94(446):590–599
Särkkä S, Sottinen T (2008) Application of Girsanov theorem to particle filtering of discretely observed continuous-time non-linear systems. Bayesian Anal 3(3):555–584
Stramer O, Yan J (2007) On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation. J Comput Graph Stat 16(3):672–691
Sun L, Lee C, Hoeting JA (2015a) Parameter inference and model selection in deterministic and stochastic dynamical models via approximate Bayesian computation: modeling a wildlife epidemic. Environmetrics 26(7):451–462
Sun L, Lee C, Hoeting JA (2015b) A penalized simulated maximum likelihood approach in parameter estimation for stochastic differential equations. Comput Stat Data Anal 84:54–67
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. EF-0914489 (Sun and Hoeting). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The research work of Chihoon Lee is supported in part by the Army Research Office under Grant No. W911NF-14-1-0216. This research also utilized the CSU ISTeC Cray HPS System, which is supported by NSF Grant CSN-0923386. We are grateful to N. Thompson Hobbs for fruitful discussions and Michael W. Miller and the Colorado Division of Parks and Wildlife for sharing the data. We also appreciate the reviewers for their insightful suggestions that have greatly enhanced the manuscript.
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Sun, L., Lee, C. & Hoeting, J.A. A penalized simulated maximum likelihood method to estimate parameters for SDEs with measurement error. Comput Stat 34, 847–863 (2019). https://doi.org/10.1007/s00180-018-0846-3
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DOI: https://doi.org/10.1007/s00180-018-0846-3