  Experimental methods based on single particle tracking (SPT) are being
increasingly employed in the physical and biological sciences, where nanoscale
objects are visualized with high temporal and spatial resolution. SPT can probe
interactions between a particle and its environment but the price to be paid is
the absence of ensemble averaging and a consequent lack of statistics. Here we
address the benchmark question of how to accurately extract the diffusion
constant of one single Brownian trajectory. We analyze a class of estimators
based on weighted functionals of the square displacement. For a certain choice
of the weight function these functionals provide the true ensemble averaged
diffusion coefficient, with a precision that increases with the trajectory
resolution.
