  Given a L\'evy process $L$, we consider the so-called statistical Skorohod
embedding problem of recovering the distribution of an independent random time
$T$ based on i.i.d. sample from $L_{T}.$ Our approach is based on the genuine
use of the Mellin and Laplace transforms. We propose a consistent estimator for
the density of $T,$ derive its convergence rates and prove their optimality. It
turns out that the convergence rates heavily depend on the decay of the Mellin
transform of $T.$ We also consider the application of our results to the
problem of statistical inference for variance-mean mixture models and for
time-changed L\'evy processes.
