  We mainly investigate the quantum measurement of Kerr nonlinearity in the
driven-dissipative system. Without the dissipation, the measurement precision
of the nonlinearity parameter $\chi$ scales as "super-Heisenberg scaling"
$1/N^2$ with $N$ being the total average number of particles (photons) due to
the nonlinear generator. Here, we find that "super-Heisenberg scaling"
$1/N^{3/2}$ can also be obtained by choosing a proper interrogation time. In
the steady state, the "super-Heisenberg scaling" $1/N^{3/2}$ can only be
achieved when the nonlinearity parameter is close to 0 in the case of the
single-photon loss and the one-photon driving or the two-photon driving. The
"super-Heisenberg scaling" disappears with the increase of the strength of the
nonlinearity. When the system suffers from the two-photon loss in addition to
the single-photon loss, the optimal measurement precision will not appear at
the nonlinearity $\chi=0$ in the case of the one-photon driving.
Counterintuitively, in the case of the two-photon driving we find that it is
not the case that the higher the two-photon loss, the lower the measurement
precision. It means that the measurement precision of $\chi$ can be improved to
some extent by increasing the two-photon loss.
