  We are interested in estimating the location of what we call "smooth
change-point" from $n$ independent observations of an inhomogeneous Poisson
process. The smooth change-point is a transition of the intensity function of
the process from one level to another which happens smoothly, but over such a
small interval, that its length $\delta\_n$ is considered to be decreasing to
$0$ as $n\to+\infty$. We show that if $\delta\_n$ goes to zero slower than
$1/n$, our model is locally asymptotically normal (with a rather unusual rate
$\sqrt{\delta\_n/n}$), and the maximum likelihood and Bayesian estimators are
consistent, asymptotically normal and asymptotically efficient. If, on the
contrary, $\delta\_n$ goes to zero faster than $1/n$, our model is non-regular
and behaves like a change-point model. More precisely, in this case we show
that the Bayesian estimators are consistent, converge at rate $1/n$, have
non-Gaussian limit distributions and are asymptotically efficient. All these
results are obtained using the likelihood ratio analysis method of Ibragimov
and Khasminskii, which equally yields the convergence of polynomial moments of
the considered estimators. However, in order to study the maximum likelihood
estimator in the case where $\delta\_n$ goes to zero faster than $1/n$, this
method cannot be applied using the usual topologies of convergence in
functional spaces. So, this study should go through the use of an alternative
topology and will be considered in a future work.
