Estimating intensity functions of spatial inhomogeneous Poisson point processes via a Stein estimator

Abstract

We develop a global Stein estimator through Malliavin derivatives to estimate the multi-variable intensity functions of general d-dimensional spatial inhomogeneous Poisson point processes (IPPPs). The gain performance between our Stein estimator and a tested maximum likelihood (ML) estimator is analyzed via Malliavin calculus. In comparison with existing homogeneous studies, our inhomogeneous cases are more complex in a global way and more real in practice. In the numerical simulation experiments, our estimated intensity functions via the Stein estimator well match the true one, which outperform the ones obtained by the ML estimator with an average gain larger than 30% concerning mean squared errors (MSEs). Hence, our Stein method is more robust with smaller MSEs. We also establish the mean-squared convergence with error estimation for our Stein estimator. Comparing with the existing study, our observation window for an IPPP is a d-dimensional rectangle and its boundary is not ${\mathcal{C}}^2$-smooth. This non-smoothness introduces additional complexity to our convergence analysis. Thus, a weak convergence approach is employed to overcome this difficulty.

Keywords:

• Multi-variable intensity function
• Spatial inhomogeneous Poisson point process
• Stein estimator
• Malliavin calculus
• Big data

Additional information

Funding

Supported by National Natural Science Foundation of China with Grant No. 11771006 and Grant No. 11371010.