Categorizing the movement of organelles, nanoparticles, and other vesicles inside living cells is an important research problem for scientists conducting fluorescence microscopy and quantum dot experiments. Particle trajectories in these complex biological systems are incredibly diverse. The relative proportions of each movement type present in a dataset can be useful to making experimental conclusions. We focus on two of the main movement patterns: free diffusion, where the particle is passively fluctuating, and anchored diffusion, which exhibits similar fluctuations but is tethered to a fixed point. We model both movement types by approximations of a stochastic differential equation and use their properties to explore rigorous model selection methods in statistics and information theory. In particular, we analyze the performance of the Akaike Information Criterion (AIC) and Bayes Factor approaches. Conducting numerical simulation allows us to determine how well these methods work at identifying each movement type. We further characterize and compare the distribution of the AIC on a given trajectory of each movement type. In cases where the true and likelihood models disagree, this presents a challenge. We derive the expression for these mismatched cases, including maximum likelihood estimators. This allows us to determine the form of a likelihood ratio test between the two models.
