  The problems of optimally estimating a phase, a direction, and the
orientation of a Cartesian frame (or trihedron) with general pure states are
addressed. Special emphasis is put on estimation schemes that allow for
inconclusive answers or abstention. It is shown that such schemes enable
drastic improvements, up to the extent of attaining the Heisenberg limit in
some cases, and the required amount of abstention is quantified. A general
mathematical framework to deal with the asymptotic limit of many qubits or
large angular momentum is introduced and used to obtain analytical results for
all the relevant cases under consideration. Parameter estimation with
abstention is also formulated as a semidefinite programming problem, for which
very efficient numerical optimization techniques exist.
