Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density

Let

Suppose that, corresponding to the true parameter pair

In this work, we will establish the posterior contraction rate in a suitable metric around the true parameter pair

The proof of our main theorem employs certain results from [

We remark that a practical implementation of the Bayesian approach to decompounding lies outside the scope of the present paper. Preliminary investigations and a small scale simulation study we performed show that it is feasible and under certain conditions leads to good results. However, the technical complications one has to deal with are quite formidable, and therefore the results of our study of implementational aspects of decompounding will be reported elsewhere.

The rest of the paper is organized as follows. In the next section, we introduce some notation and recall a number of notions useful for our purposes. Section

Assume without loss of generality that

law of

law of

law of

We will first specify the dominating measure for

We will use the product prior

The prior for

Let

The Hellinger distance

For any

Let

Define the complements of the Hellinger-type neighborhoods of

The

In order to derive the posterior contraction rate in our problem, we impose the following conditions on the true parameter pair

Denote by

The true density

The conditions on

We also need to make some assumptions on the prior

The prior

The prior

The base measure

There exist strictly positive constants

This assumption comes from [

Assumption (

We now state our main result.

We conclude this section with a brief discussion on the obtained result: the logarithmic factor

After completion of this work, we learned about the paper [

The proof of Theorem

The proof of the lemma is given in Section

Let

In [

Introduce the notation

We first focus on (

For the verification of (

We have thus verified conditions (

We start with a lemma from [

The proof of the lemma consists in an application of Jensen’s inequality for conditional expectations. This lemma is typically used as follows. The measures

In the proof of Lemma

Application of Lemma

We have

Combining the estimates on I and

First, note that for

The authors would like to thank the referee for his/her remarks. The research leading to these results has received funding from the European Research Council under ERC Grant Agreement 320637.