We revisit the problem of estimating the intensity parameter of a homogeneous Poisson point process observed in a bounded window of Rd making use of a (now) old idea of James and Stein. For this, we prove an integration by parts formula for functionals defined on the Poisson space. This formula extends the one obtained by Privault and Réveillac (Statistical inference for Stochastic Processes, 2009) in the one-dimensional case and is well-suited to a notion of derivative of Poisson functionals which satisfy the chain rule. The new estimators can be viewed as biased versions of the MLE with a tailored-made bias designed to reduce the variance of the MLE. We study a large class of examples and show that with a controlled probability the corresponding estimator outperforms the MLE. We illustrate in a simulation study that for very reasonable practical cases (like an intensity of 10 or 20 of a Poisson point process observed in the euclidean ball of dimension between 1 and 5) we can obtain a relative (mean squared error) gain above 20% for the Stein estimator with respect to the maximum likelihood.

This is a joint work with M. Clausel and J. Lelong (Univ. Grenoble Alpes).