A non-parametric Bayesian approach to decompounding from high frequency data

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density $f_0$ of its jump sizes, as well as of its intensity $\lambda_0.$ We take a Bayesian approach to the problem and specify the prior on $f_0$ as the Dirichlet location mixture of normal densities. An independent prior for $\lambda_0$ is assumed to be compactly supported and possess a positive density with respect to the Lebesgue measure. We show that under suitable assumptions the posterior contracts around the pair $(\lambda_0,f_0)$ at essentially (up to a logarithmic factor) the $\sqrt{n\Delta}$-rate, where $n$ is the number of observations and $\Delta$ is the mesh size at which the process is sampled. The emphasis is on high frequency data, $\Delta\to 0$, but the obtained results are also valid for fixed $\Delta$. In either case we assume that $n\Delta\rightarrow\infty$. Our main result implies existence of Bayesian point estimates converging (in the frequentist sense, in probability) to $(\lambda_0,f_0)$ at the same rate. Simulations complement the theory.