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In this paper, we consider the problem of the accuracy of estimating the location and other attributes of a moving single molecule whose trajectory is acquired in a sequence of time intervals by a pixelated detector. We present expressions of the Fisher information matrices from which the benchmark for the accuracy of the parameter estimates is obtained. In the absence of extraneous noise, it is shown that time discretization of the image acquisition process results in a limit of the accuracy of the parameter estimates that is better than or at least as good as that acquired without time discretization. This analytical result is also illustrated by simulations. However, in the presence of extraneous noise, simulations show that finer time discretization may not always lead to better limit of the accuracy than that acquired without time discretization of the image acquisition process.