  We consider an inhomogeneous Poisson process $X$ on $[0,T]$. The intensity
function of $X$ is supposed to be strictly positive and smooth on $[0,T]$
except at the point $\theta$, in which it has either a 0-type singularity
(tends to 0 like $\abs{x}^p$, $p\in(0,1)$), or an $\infty$-type singularity
(tends to $\infty$ like $\abs{x}^p$, $p\in(-1,0)$). We suppose that we know the
shape of the intensity function, but not the location of the singularity. We
consider the problem of estimation of this location (shift) parameter $\theta$
based on $n$ observations of the process $X$. We study the Bayesian estimators
and, in the case $p>0$, the maximum likelihood estimator. We show that these
estimators are consistent, their rate of convergence is $n^{1/(p+1)}$, they
have different limit distributions, and the Bayesian estimators are
asymptotically efficient.
