Modern Stochastics: Theory and Applications

Let $N=(N_{t})_{t\ge 0}$ be a Poisson process of constant intensity $\lambda >0$, and let $\{Y_{j}\}$ be independent and identically distributed (i.i.d.) ${\mathbb{R}}^{d}$-valued random vectors defined on the same probability space and having a common distribution function R, which is assumed to be absolutely continuous with respect to the Lebesgue measure with density r. Assume that N and $\{Y_{j}\}$ are independent and define the ${\mathbb{R}}^{d}$-valued process $X=(X_{t})_{t\ge 0}$ by The process X is called a compound Poisson process (CPP) and forms a basic stochastic model in a variety of applied fields, such as, for example, risk theory and queueing; see [10, 21].

Suppose that, corresponding to the true parameter pair $(\lambda _{0},r_{0})$, a sample $X_{\varDelta }$, $X_{2\varDelta },\dots ,X_{n\varDelta }$ from X is available, where the sampling mesh $\varDelta >0$ is assumed to be fixed and thus independent of n. The problem we study in this note is nonparametric estimation of $r_{0}$ (and of $\lambda _{0}$). This is referred to as decompounding and is well studied for one-dimensional CPPs; see [2, 3, 6, 9, 24]. Some practical situations in which this problem may arise are listed in [9, p. 3964]. However, the methods used in the above papers do not seem to admit (with the exception of [24]) a generalization to the multidimensional setup. This is also true for papers studying nonparametric inference for more general classes of Lévy processes (of which CPPs form a particular class), such as, for example, [4, 5, 19]. In fact, there is a dearth of publications dealing with nonparametric inference for multidimensional Lévy processes. An exception is [1], where the setup is however specific in that it is geared to inference in Lévy copula models and that, unlike the present work, the high-frequency sampling scheme is assumed ($\varDelta =\varDelta _{n}\to 0$ and $n\varDelta _{n}\to \infty $).

In this work, we will establish the posterior contraction rate in a suitable metric around the true parameter pair $(\lambda _{0},r_{0})$. This concerns study of asymptotic frequentist properties of Bayesian procedures, which has lately received considerable attention in the literature (see, e.g., [14, 15]), and is useful in that it provides their justification from the frequentist point of view. Our main result says that for a β-Hölder regular density $r_{0}$, under some suitable additional assumptions on the model and the prior, the posterior contracts at the rate ${n}^{-\beta /(2\beta +d)}{(\log n)}^{\ell }$, which, perhaps up to a logarithmic factor, is arguably the optimal posterior contraction rate in our problem. Finally, our Bayesian procedure is adaptive: the construction of our prior does not require knowledge of the smoothness level β in order to achieve the posterior contraction rate given above.

The proof of our main theorem employs certain results from [14, 22] but involves a substantial number of technicalities specifically characteristic of decompounding.

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 3 contains our main result, Theorem 2, and a brief discussion on it. The proof of Theorem 2 is given in Section 4. Finally, Section 5 contains the proof of the key technical lemma used in our proofs.

Assume without loss of generality that $\varDelta =1$, and let $Z_{i}=X_{i}-X_{i-1}$, $i=1,\dots ,n$. The ${\mathbb{R}}^{d}$-valued random vectors $Z_{i}$ are i.i.d. copies of a random vector where $\{Y_{j}\}$ are i.i.d. with distribution function $R_{0}$, whereas T, which is independent of $\{Y_{j}\}$, has the Poisson distribution with parameter $\lambda _{0}$. The problem of decompounding the jump size density $r_{0}$ introduced in Section 1 is equivalent to estimation of $r_{0}$ from observations $\mathcal{Z}_{n}=\{Z_{1},Z_{2},\dots ,Z_{n}\}$, and we will henceforth concentrate on this alternative formulation. We will use the following notation:

Define the complements of the Hellinger-type neighborhoods of $(\lambda _{0},r_{0})$ by where $\{\varepsilon _{n}\}$ is a sequence of positive numbers. We say that $\varepsilon _{n}$ is a posterior contraction rate if there exists a constant $M>0$ such that as $n\to \infty $ in ${\mathbb{Q}_{\lambda _{0},r_{0}}^{n}}$-probability.

The ε-covering number of a subset B of a metric space equipped with the metric ρ is the minimum number of ρ-balls of radius ε needed to cover it. Let $\mathcal{Q}$ be a set of CPP laws $\mathbb{Q}_{\lambda ,r}$. Furthermore, we set We recall the following general result on posterior contraction rates.

In order to derive the posterior contraction rate in our problem, we impose the following conditions on the true parameter pair $(\lambda _{0},r_{0})$.

The conditions on $r_{0}$ come from Theorem 1 in [22] and are quite reasonable. They simplify greatly when $r_{0}$ has a compact support.

We also need to make some assumptions on the prior Π defined in Section 2.2.

This assumption comes from [22, p. 626], to which we refer for an additional discussion. In particular, it is shown there that an inverse Wishart distribution (a popular prior distribution for covariance matrices) satisfies the assumptions on G with $\kappa =2$. As far as α is concerned, we can take it such that its rescaled version $\overline{\alpha }$ is a nondegenerate Gaussian distribution on ${\mathbb{R}}^{d}$.

We now state our main result.

We conclude this section with a brief discussion on the obtained result: the logarithmic factor ${(\log n)}^{\ell }$ is negligible for practical purposes. If $\kappa =1$, then the posterior contraction rate obtained in Theorem 2 is essentially ${n}^{-2\beta /(2\beta +d)}$, which is the minimax estimation rate in a number of nonparametric settings. This is arguably also the minimax estimation rate in our problem as well (cf. Theorem 2.1 in [16] for a related result in the one-dimensional setting), although here we do not give a formal argument. Equally important is the fact that our result is adaptive: the posterior contraction rate in Theorem 2 is attained without the knowledge of the smoothness level β being incorporated in the construction of our prior Π. Finally, Theorem 2, in combination with Theorem 2.5 and the arguments on pp. 506–507 in [15], implies the existence of Bayesian point estimates achieving (in the frequentist sense) this convergence rate.

The proof of Theorem 2 consists in verification of the conditions in Theorem 1. The following lemma plays the key role.

The proof of the lemma is given in Section 5. We proceed with the proof of Theorem 2.

Let $\varepsilon _{n}={n}^{-\gamma }{(\log n)}^{\ell }$ for γ and $\ell >\ell _{0}$ as in the statement of Theorem 2. Set $\overline{\varepsilon }_{n}=2\overline{C}\varepsilon _{n}$, where $\overline{C}$ is the constant from Lemma 1. We define the sieves of densities $\mathcal{F}_{n}$ as in Theorem 5 in [22]: where and $a_{1}$ and $a_{2}$ are as in Assumption 2. We also put

In [22], sieves of the type $\mathcal{F}_{n}$ are used to verify conditions of Theorem 1 and to determine posterior contraction rates in the standard density estimation context. We further will show that these sieves also work in the case of decompounding by verifying the conditions of Theorem 1 for the sieves $\mathcal{Q}_{n}$ defined in (17).

We start with a lemma from [7], which will be used three times in the proof of Lemma 1. Consider a probability space $(\varOmega ,\mathfrak{F},\mathbb{P})$. Let $\mathbb{P}_{0}$ be a probability measure on $(\varOmega ,\mathfrak{F})$ and assume that $\mathbb{P}_{0}\ll \mathbb{P}$ with Radon–Nikodym derivative $\zeta =\frac{\mathrm{d}\mathbb{P}_{0}}{\mathrm{d}\mathbb{P}}$. Furthermore, let $\mathfrak{G}$ be a sub-σ-algebra of $\mathfrak{F}$. The restrictions of $\mathbb{P}$ and $\mathbb{P}_{0}$ to $\mathfrak{G}$ are denoted ${\mathbb{P}^{\prime }}$ and ${\mathbb{P}^{\prime }_{0}}$, respectively. Then ${\mathbb{P}^{\prime }_{0}}\ll {\mathbb{P}^{\prime }}$ and $\frac{\mathrm{d}{\mathbb{P}^{\prime }_{0}}}{\mathrm{d}{\mathbb{P}^{\prime }}}=\mathbb{E}_{\mathbb{P}}[\zeta |\mathfrak{G}]=:{\zeta ^{\prime }}$.

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 $\mathbb{P}$ and $\mathbb{P}_{0}$ are possible distributions of some random element X. If ${X^{\prime }}=T(X)$ is some measurable transformation of X, then we consider ${\mathbb{P}^{\prime }}$ and ${\mathbb{P}^{\prime }_{0}}$ as the corresponding distributions of ${X^{\prime }}$. Here T may be a projection. In the present context, we take $X=(X_{t},t\in [0,1])$ and ${X^{\prime }}=X_{1}$, and so $\mathbb{P}$ in the lemma should be taken as $\mathbb{R}=\mathbb{R}_{\lambda ,r}$ and ${\mathbb{P}^{\prime }}$ as $\mathbb{Q}=\mathbb{Q}_{\lambda ,r}$.

In the proof of Lemma 1, for economy of notation, a constant $c(\underline{\lambda },\overline{\lambda })$ depending on $\underline{\lambda }$ and $\overline{\lambda }$ may differ from line to line. We also abbreviate $\mathbb{Q}_{\lambda _{0},r_{0}}$ and $\mathbb{Q}_{\lambda ,r}$ to $\mathbb{Q}_{0}$ and $\mathbb{Q}$, respectively. The same convention will be used for $\mathbb{R}_{\lambda _{0},r_{0}}$, $\mathbb{R}_{\lambda ,r}$, $\mathbb{P}_{r_{0}}$, and $\mathbb{P}_{r}$.