Optimal statistical inference for subdiffusion processes

Abstract

Statistical inference for subdiffusion processes is a growing topic of research spurred on by the advent single particle tracking technologies which allow for direct observation of the trajectories of many particles at a time. Often statistical inference is performed using methods derived from the non-ergodicity properties of subdiffusion processes. However, these estimators require in-principle continuous sampling of many long trajectories which is not implementable without introducing some discretisation error, and once the discretisation is performed, the process is more data intensive than methods which require only a single data point per trajectory. Therefore, we consider as an alternative the classical approach to statistical inference using a single long or short increment as data, which is free from these drawbacks. We investigate the Fisher information for the diffusion coefficient and the stability index in order to quantify how much information is available when sampling in these regimes. We observe that the Fisher information for the respective parameters is the same regardless of whether they sampled in the short or long trajectory regime, implying that in the best case scenario a great deal of experimental expense could be saved without the loss of any statistical information regarding the parameters by observing only short increments rather than long ones.