  We present a parameter estimation technique based on performing joint
measurements of a weak interaction away from the weak-value-amplification
approximation. Two detectors are used to collect full statistics of the
correlations between two weakly entangled degrees of freedom. Without the need
of postselection, the protocol resembles the anomalous amplification of an
imaginary-weak-value-like response. The amplification is induced in the
difference signal of both detectors allowing robustness to different sources of
technical noise, and offering in addition the advantages of balanced signals
for precision metrology. All of the Fisher information about the parameter of
interest is collected, and a phase controls the amplification response. We
experimentally demonstrate the proposed technique by measuring polarization
rotations in a linearly polarized laser pulse. The effective sensitivity and
precision of a split detector is increased when compared to a conventional
continuous-wave balanced detection technique.
