  We derive the class of covariant measurements which are optimal according to
the maximum likelihood criterion. The optimization problem is fully resolved in
the case of pure input states, under the physically meaningful hypotheses of
unimodularity of the covariance group and measurability of the stability
subgroup. The general result is applied to the case of covariant state
estimation for finite dimension, and to the Weyl-Heisenberg displacement
estimation in infinite dimension. We also consider estimation with multiple
copies, and compare collective measurements on identical copies with the scheme
of independent measurements on each copy. A "continuous-variables" analogue of
the measurement of direction of the angular momentum with two anti-parallel
spins by Gisin and Popescu is given.
