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A variety of researchers have successfully obtained the parameters of lowdimensional diffusion models using the data that comes out of atomisticsimulations. This naturally raises a variety of questions about efficientestimation, goodness-of-fit tests, and confidence interval estimation. Thefirst part of this article uses maximum likelihood estimation to obtain theparameters of a diffusion model from a scalar time series. I address numericalissues associated with attempting to realize asymptotic statistics results withmoderate sample sizes in the presence of exact and approximated transitiondensities. Approximate transition densities are used because the analyticsolution of a transition density associated with a parametric diffusion modelis often unknown.I am primarily interested in how well the deterministictransition density expansions of Ait-Sahalia capture the curvature of thetransition density in (idealized) situations that occur when one carries outsimulations in the presence of a "glassy" interaction potential. Accurateapproximation of the curvature of the transition density is desirable becauseit can be used to quantify the goodness-of-fit of the model and to calculateasymptotic confidence intervals of the estimated parameters. The second part ofthis paper contributes a heuristic estimation technique for approximating anonlinear diffusion model. A "global" nonlinear model is obtained by taking abatch of time series and applying simple local models to portions of the data.I demonstrate the technique on a diffusion model with a known transitiondensity and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly  shortened

Fitting Effective Diffusion Models to Data Associated with a “Glassy" Potential: Estimation, Classical Inference Procedures, and Some Heuristics