  Postselected weak measurement is a useful protocol for amplifying weak
physical effects. However, there has recently been controversy over whether it
gives any advantage in precision. While it is now clear that retaining failed
postselections can yield more Fisher information than discarding them, the
advantage of postselection measurement itself still remains to be clarified. In
this Letter, we address this problem by studying two widely used estimation
strategies: averaging measurement results, and maximum likelihood estimation,
respectively. For the first strategy, we find a surprising result that squeezed
coherent states of the pointer can give postselected weak measurements a higher
signal-to-noise ratio than standard ones while all standard coherent states
cannot, which suggests that raising the precision of weak measurements by
postselection calls for the presence of "nonclassicality" in the pointer
states. For the second strategy, we show that the quantum Fisher information of
postselected weak measurements is generally larger than that of standard weak
measurements, even without using the failed postselection events, but the gap
can be closed with a proper choice of system state.
