  The main goal of quantum metrology is to obtain accurate values of physical
parameters using quantum probes. In this context, we show that abstention,
i.e., the possibility of getting an inconclusive answer at readout, can
drastically improve the measurement precision. We focus on phase estimation and
quantify the required amount of abstention for a given precision. We also
develop analytical tools to obtain the asymptotic behavior of the precision and
required rate of abstention for arbitrary pure qubit states.
