  We present a detailed methodological study of the application of the modified
profile likelihood method for the calibration of nonlinear financial models
characterised by a large number of parameters. We apply the general approach to
the Log-Periodic Power Law Singularity (LPPLS) model of financial bubbles. This
model is particularly relevant because one of its parameters, the critical time
$t_c$ signalling the burst of the bubble, is arguably the target of choice for
dynamical risk management. However, previous calibrations of the LPPLS model
have shown that the estimation of $t_c$ is in general quite unstable. Here, we
provide a rigorous likelihood inference approach to determine $t_c$, which
takes into account the impact of the other nonlinear (so-called "nuisance")
parameters for the correct adjustment of the uncertainty on $t_c$. This
provides a rigorous interval estimation for the critical time, rather than a
point estimation in previous approaches. As a bonus, the interval estimations
can also be obtained for the nuisance parameters ($m,\omega$, damping), which
can be used to improve filtering of the calibration results. We show that the
use of the modified profile likelihood method dramatically reduces the number
of local extrema by constructing much simpler smoother log-likelihood
landscapes. The remaining distinct solutions can be interpreted as genuine
scenarios that unfold as the time of the analysis flows, which can be compared
directly via their likelihood ratio. Finally, we develop a multi-scale profile
likelihood analysis to visualize the structure of the financial data at
different scales (typically from 100 to 750 days). We test the methodology
successfully on synthetic price time series and on three well-known historical
financial bubbles.
