  Let $X=(X_t)_{t\geq 0}$ be a known process and $T$ an unknown random time
independent of $X$. Our goal is to derive the distribution of $T$ based on an
iid sample of $X_T$. Belomestny and Schoenmakers (2015) propose a solution
based the Mellin transform in case where $X$ is a Brownian motion. Applying
their technique we construct a non-parametric estimator for the density of $T$
for a self-similar one-dimensional process $X$. We calculate the minimax
convergence rate of our estimator in some examples with a particular focus on
Bessel processes where we also show asymptotic normality.
