  The central problem of quantum statistics is to devise measurement schemes
for the estimation of an unknown state, given an ensemble of $n$ independent
identically prepared systems. For locally quadratic loss functions, the risk of
standard procedures has the usual scaling of $1/n$. However, it has been
noticed that for fidelity based metrics such as the Bures distance, the risk of
conventional (non-adaptive) qubit tomography schemes scales as $1/\sqrt{n}$ for
states close to the boundary of the Bloch sphere. Several proposed estimators
appear to improve this scaling, and our goal is to analyse the problem from the
perspective of the maximum risk over all states.
  We propose qubit estimation strategies based on separate and adaptive
measurements, that achieve $1/n$ scalings for the maximum Bures risk. The
estimator involving local measurements uses a fixed fraction of the available
resource $n$ to estimate the Bloch vector direction; the length of the Bloch
vector is then estimated from the remaining copies by measuring in the
estimator eigenbasis. The estimator based on collective measurements uses local
asymptotic normality techniques which allows us derive upper and lower bounds
to its maximum Bures risk. We also discuss how to construct a minimax optimal
estimator in this setup. Finally, we consider quantum relative entropy and show
that the risk of the estimator based on collective measurements achieves a rate
$O(n^{-1}\log n)$ under this loss function. Furthermore, we show that no
estimator can achieve faster rates, in particular the `standard' rate $1/n$.
