  The movement of a particle described by Brownian motion is quantified by a
single parameter, $D$, the diffusion constant. The estimation of $D$ from a
discrete sequence of noisy observations is a fundamental problem in biological
single particle tracking experiments since it can report on the environment
and/or the state of the particle itself via hydrodynamic radius. Here we
present a method to estimate $D$ that takes into account several effects that
occur in practice, that are important for correct estimation of $D$, and that
have hitherto not been combined together for estimation of $D$. These effects
are motion blur from finite integration time of the camera, intermittent
trajectories, and time-dependent localization uncertainty. Our estimation
procedure, a maximum likelihood estimation, follows directly from the
likelihood expression for a discretely observed Brownian trajectory that
explicitly includes these effects. The manuscript begins with the formulation
of the likelihood expression and then presents three methods to find the exact
solution. Each method has its own advantages in either computational
robustness, theoretical insight, or the estimation of hidden variables. We then
compare our estimator to previously published estimators using a squared log
loss function to demonstrate the benefit of including these effects.
