  As quantum tomography is becoming a key component of the quantum engineering
toolbox, there is a need for a deeper understanding of the multitude of
estimation methods available. Here we investigate and compare several such
methods: maximum likelihood, least squares, generalised least squares, positive
least squares, thresholded least squares and projected least squares. The
common thread of the analysis is that each estimator projects the measurement
data onto a parameter space with respect to a specific metric, thus allowing us
to study the relationships between different estimators.
  The asymptotic behaviour of the least squares and the projected least squares
estimators is studied in detail for the case of the covariant measurement and a
family of states of varying ranks. This gives insight into the rank-dependent
risk reduction for the projected estimator, and uncovers an interesting
non-monotonic behaviour of the Bures risk. These asymptotic results complement
recent non-asymptotic concentration bounds of \cite{GutaKahnKungTropp} which
point to strong optimality properties, and high computational efficiency of the
projected linear estimators.
  To illustrate the theoretical methods we present results of an extensive
simulation study. An app running the different estimators has been made
available online.
