Efficient and superefficient estimators of filtered Poisson process intensities

ABSTRACT

Let N^K = {N^K_t, t ∈ [0, T]} be a filtered Poisson process defined on a probability space $(\Omega ,\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\in [0,T]},P)$, and let θ ≔ (θ_t, t ∈ [0, T]) be a deterministic function which is the intensity of N^K under a probability P_θ. In the present paper we prove that the natural maximum likelihood estimator (MLE) N^K is an efficient estimator for θ under P_θ. Using Malliavin calculus we construct superefficient estimators of Stein type for θ which dominate, under the usual quadratic risk, the MLE N^K. These superefficient estimators are given under the form $N_t^K+D_t^{{\tilde{N}}^K}log\left(F\right)$ where F is a random variable satisfying some assumptions and $D_t^{{\tilde{N}}^K}$ is the Malliavin derivative with respect to the compensated version ${\tilde{N}}^K$ of N^K.

KEYWORDS:

• Filtered Poisson process
• Intensity estimation
• Stein estimation
• Malliavin calculus.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05
• 60G55
• 60H07

Acknowledgements

The authors would like to thank the referee for several helpful corrections and suggestions that led to many improvements in the paper.