  In this paper, we deal with the problem of calibrating thresholding rules in
the setting of Poisson intensity estimation. By using sharp concentration
inequalities, oracle inequalities are derived and we establish the optimality
of our estimate up to a logarithmic term. This result is proved under mild
assumptions and we do not impose any condition on the support of the signal to
be estimated. Our procedure is based on data-driven thresholds. As usual, they
depend on a threshold parameter $\gamma$ whose optimal value is hard to
estimate from the data. Our main concern is to provide some theoretical and
numerical results to handle this issue. In particular, we establish the
existence of a minimal threshold parameter from the theoretical point of view:
taking $\gamma<1$ deteriorates oracle performances of our procedure. In the
same spirit, we establish the existence of a maximal threshold parameter and
our theoretical results point out the optimal range $\gamma\in[1,12]$. Then, we
lead a numerical study that shows that choosing $\gamma$ larger than 1 but
close to 1 is a fairly good choice. Finally, we compare our procedure with
classical ones revealing the harmful role of the support of functions when
estimated by classical procedures.
