  The weak-value (WV) measurement proposed by Aharonov, Albert and Vaidman
(AAV) has attracted a great deal of interest in connection with quantum
metrology. In this work, we extend the analysis beyond the AAV limit and obtain
a few main results. (i) We obtain non-perturbative result for the
signal-to-noise ratio (SNR). In contrast to the AAV's prediction, we find that
the SNR asymptotically gets worse when the AAV's WV $A_w$ becomes large, i.e.,
in the case $g|A_w|^2>>1$, where $g$ is the measurement strength. (ii) With the
increase of $g$ (but also small), we find that the SNR is comparable to the
result under the AAV limit, while both can reach -- actually the former can
slightly exceed -- the SNR of the standard measurement. However, along a
further increase of $g$, the WV technique will become less efficient than the
standard measurement, despite that the postselection probability is increased.
(iii) We find that the Fisher information can characterize the estimate
precision qualitatively well as the SNR, yet their difference will become more
prominent with the increase of $g$. (iv) We carry out analytic expressions of
the SNR in the presence of technical noises and illustrate the particular
advantage of the imaginary WV measurement. The non-perturbative result of the
SNR manifests a favorable range of the noise strength and allows an optimal
determination.
