Nonparametric Bayesian inference for multidimensional compound Poisson processes

Abstract

Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity ${\lambda }_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,{\lambda }_0)$ at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional Lévy processes, and the study of their rates of convergence is a new contribution to the literature.

Published at http://dx.doi.org/10.15559/15-VMSTA20 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

References in corpus (4)

• Central limit theorems for sequences of multiple stochastic integrals
• Adaptive Bayesian multivariate density estimation with Dirichlet mixtures
• A kernel type nonparametric density estimator for decompounding
• Nonparametric inference on Lévy measures and copulas

Cited by in corpus (7)

• Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes
• Efficient nonparametric inference for discretely observed compound Poisson processes
• A non-parametric Bayesian approach to decompounding from high frequency data
• Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme
• Bayesian non-parametric inference for $\Lambda$-coalescents: consistency and a parametric method
• Decompounding discrete distributions: A non-parametric Bayesian approach
• Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations