Abstract
Given a sample from a discretely observed compound Poisson process, we consider nonparametric 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 to 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, \(\Delta \rightarrow 0,\) 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. We also discuss a practical implementation of our approach. The computational problem is dealt with by inclusion of auxiliary variables and we develop a Markov chain Monte Carlo algorithm that samples from the joint distribution of the unknown parameters in the mixture density and the introduced auxiliary variables. Numerical examples illustrate the feasibility of this approach.
Introduction
Problem formulation
Let \(N=(N_{t},\, t\ge 0)\) be a Poisson process with a constant intensity \(\lambda >0\) and let \(Y_1,\,Y_2,\,Y_3,\ldots \) be a sequence of independent random variables independent of N and having a common distribution function F with density f (with respect to the Lebesgue measure). A compound Poisson process (abbreviated CPP) \(X=(X_t,\, t\ge 0)\) is defined as
where the sum over an empty set is by definition equal to zero. CPPs form a basic model in a variety of applied fields, most notably in e.g., queueing and risk theory, see Embrechts et al. (1997) and Prabhu (1998) and the references therein, but also in other fields of science, see, e.g., Alexandersson (1985) and Burlando and Rosso (1993) for stochastic models for precipitation, Katz (2002) on modelling of hurricane damage, or Scalas (2006) for applications in economics and finance.
Suppose that corresponding to the ‘true’ parameter values \(\lambda =\lambda _0\) and \(f=f_0,\) a discrete time sample \(X_{\Delta },\,X_{2\Delta },\ldots ,X_{n\Delta }\) is available from (1), where \(\Delta >0.\) Such a discrete time observation scheme is common in a number of applications of CPP, e.g., in the precipitation models of the above references. Based on the sample \(\mathcal {X}_n^\Delta =(X_\Delta ,\,X_{2\Delta },\ldots ,X_{n\Delta }),\) we are interested in (nonparametric) estimation of \(\lambda _0\) and \(f_0.\) Before proceeding further, we notice that by the stationary independent increments property of a CPP, the random variables \(Z_i^{\Delta } = X_{i\Delta }X_{(i1)\Delta },\,1\le i \le n,\) are independent and identically distributed. Each \(Z_i^{\Delta }\) has the same distribution as the random variable
where \(T^{\Delta }\) is independent of the sequence \(Y_1,\, Y_2,\ldots \) and has a Poisson distribution with parameter \(\Delta \lambda .\) Hence, our problem is equivalent to estimating (nonparametrically) \(\lambda _0\) and \(f_0\) based on the sample \(\mathcal {Z}_n^{\Delta }=(Z_1^{\Delta },\,Z_2^{\Delta },\ldots ,Z_n^{\Delta }).\) We will henceforth use this alternative formulation of the problem. Our emphasis is on high frequency data, \(\Delta =\Delta _n\rightarrow 0\) as \(n\rightarrow \infty ,\) but the obtained results are also valid for low frequency observations, i.e., for fixed \(\Delta .\)
Our main result is on the contraction rate of the posterior distribution, which we show to be, up to a logarithmic factor, \((n\Delta )^{1/2}.\) A by now standard approach to obtain contraction rates in an IID setting is to verify the assumptions of the fundamental Theorem 2.1 in Ghosal et al. (2000). It should be noted that in the present high frequency setting, this theorem is not applicable. One of the model assumptions underlying this theorem, which is satisfied in Gugushvili et al. (2015), is that one deals with samples of a fixed distribution, whereas in our present high frequency observation regime the distribution of \(Z^\Delta \) is varying, with the Dirac distribution concentrated at zero as its limit for \(\Delta \rightarrow 0.\) Therefore we propose an alternative approach, circumventing the use of the cited Theorem 2.1. The theoretical contribution of the present paper is therefore not only the statement of the main result itself, but also its proof. Next to this we also discuss a practical implementation of our nonparametric Bayesian approach, a Markov chain Monte Carlo algorithm that samples from the joint distribution of the unknown parameters in the mixture density and certain introduced auxiliary variables.
Literature review and present approach
Because adding a Poisson number of \(Y_j\)’s amounts to compounding their distributions, the problem of recovering the intensity \(\lambda _0\) and the density \(f_0\) from the observations \(Z_i\)’s can be referred to as decompounding. Decompounding already has some history: the early contributions (Buchmann and Grübel 2003, 2004) dealt with estimation of the distribution function \(F_0,\) paying particular attention to the case when \(F_0\) is discrete, while the later contributions (Comte et al. 2014; Duval 2013; Es et al. 2007) concentrated on estimation of the density \(f_0\) instead. More (frequentist) theory on statistical inference on CPPs (and more generally on Lévy processes) can be found in the volume (Belomestny et al. 2015), with the survey paper (Comte et al. 2015) devoted to statistical methods for high frequency discrete observations, with a special section on CPPs. Other references on statistics for Lévy processes in the high frequency data setting are Comte and GenonCatalot (2011), Comte and GenonCatalot (2010), Comte et al. (2010), FigueroaLópez (2008), FigueroaLopez (2009), Nickl and Reiß (2012), Nickl et al. (2016), and Ueltzhöfer and Klüppelberg (2011). All these approaches are frequentist in nature. On the other hand, theoretical and computational advances made over the recent years have shown that a nonparametric Bayesian approach is feasible in various statistical settings; see e.g., Hjort et al. (2010) for an overview. This is the approach we will take in this work to estimate \(\lambda _0\) and \(f_0.\)
To the best of our knowledge, nonparametric Bayesian approach to inference for (a class of) Lévy processes was first considered in Gugushvili et al. (2015). That paper, contrary to the present context, dealt with observations at fixed equidistant times, and was strongly based on an application of Theorem 2.1 of Ghosal et al. (2000), as already alluded to in the problem formulation of Sect. 1.1. The present work complements the results from Gugushvili et al. (2015), in the sense that we now allow high frequency observations, which requires a substantially different route to prove our results, as we will explain in more detail in Sect. 1.3.
We will study the nonparametric Bayesian approach to decompounding from a frequentist point of view (in the sense specified below), so that one may also think of it as a means for obtaining a frequentist estimator. Advantages of the nonparametric Bayesian approach include automatic quantification of uncertainty in parameter estimates through Bayesian posterior credible sets and automatic selection of the degree of smoothing required in nonparametric inferential procedures.
Results
The nonparametric class \(\mathcal {F}\) of densities f that we consider is that of location mixtures of normal densities. So we consider densities specified by
where \(\phi _{\sigma }\) denotes the density of the normal distribution with mean zero and variance \(\sigma ^2\) and H is a mixing measure. These mixtures form a rich and flexible class of densities, see Marron and Wand (1992) and McLachlan and Peel (2000), that are capable of closely approximating many densities that themselves are not representable in this way. The resulting mixture densities will be infinitely smooth, which is arguably the case in many, if not most, practical applications.
Bayesian estimation requires specification of prior distributions on \(\lambda \) and f. We propose independent priors on \(\lambda \) and f that we denote by \(\Pi _1\) and \(\Pi _2,\) respectively. For f, we take a Dirichlet mixture of normal densities as a prior. This type of prior in the context of Bayesian density estimation has been introduced in Ferguson (1983) and Lo (1984); for recent references see, e.g., Ghosal and Vaart (2001). The prior for f is defined as the law of the function \(f_{ H , \sigma }\) as in (3), with H assumed to follow a Dirichlet process prior \(D_\alpha \) with base measure \(\alpha \) and \(\sigma \) a priori independent with distribution \(\Pi _3.\) Recall that a Dirichlet process \(D_{\alpha }\) on \(\mathbb {R}\) with the base measure \(\alpha \) defined on the Borel \(\sigma \)algebra \(\mathcal {B}(\mathbb {R})\) (we assume \(\alpha \) to be nonnegative and \(\sigma \)additive) is a random probability measure G on \(\mathbb {R},\) such that for every finite and measurable partition \(B_1,\,B_2,\ldots ,B_k\) of \(\mathbb {R},\) the probability vector \((G(B_1),\,G(B_2),\ldots ,G(B_k))\) possesses the Dirichlet distribution on the kdimensional simplex with parameters \((\alpha (B_1),\,\alpha (B_2),\ldots ,\alpha (B_k)).\) See, e.g., the original paper (Ferguson 1973), or the overview article (Ghosal 2010) for more information on Dirichlet process priors.
A nonparametric Bayesian approach to density estimation employing a Dirichlet mixture of normal densities as a prior can in very rough sense be thought of as a Bayesian counterpart of kernel density estimation (with a Gaussian kernel), cf. Ghosal and van der Vaart (2007, p. 697).
With the sample size n tending to infinity, the Bayesian approach should be able to discern the true parameter pair \((\lambda _0,\,f_0)\) with increasing accuracy. We can formalise this by requiring, for instance, that for any fixed neighbourhood A (in an appropriate topology) of \((\lambda _0,\,f_0),\, \Pi (A^c\mathcal {Z}_{n}^\Delta )\rightarrow 0\) in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability. Here \(\Pi \) is used as a shorthand notation for the posterior distribution of \((\lambda ,\,f)\) and we use \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta }\) to denote the law of the random variable \(Z^{\Delta }\) in (2) and \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\) the law of \(\mathcal {Z}_n^{\Delta }.\) More generally, one may take a sequence of shrinking neighbourhoods \(A_n\) of \((\lambda _0,\,f_0)\) and try to determine the rate at which the neighbourhoods \(A_n\) are allowed to shrink, while still capturing most of the posterior mass. This rate is referred to as a posterior convergence rate (we will give the precise definition in Sect. 3). Two fundamental references dealing with establishing it in various statistical settings are Ghosal et al. (2000) and Ghosal and Vaart (2001). This convergence rate can be thought of as an analogue of the convergence rate of a frequentist estimator. The analogy can be made precise: contraction of the posterior distribution at a certain rate implies existence of a Bayes point estimate with the same convergence rate (in the frequentist sense); see Theorem 2.5 in Ghosal et al. (2000) and the discussion on pp. 506–507 there.
Obviously, for our programme to be successful, \(\Delta \) has to satisfy the assumption \(n\Delta \rightarrow \infty ,\) which is a necessary condition for consistent estimation of \((\lambda _0,\,f_0),\) as it ensures that asymptotically we observe an infinite number of jumps in the process. We cover both the case of so called high frequency observation schemes (\(\Delta \rightarrow 0\)) as well as low frequency observations (fixed \(\Delta \)). A sufficient condition, which covers both observation regimes and which relates \(\Delta \) to n, is \(\Delta =n^{\alpha },\) where \(0\le \alpha <1.\)
We note that in Ghosal and Tang (2006) and Tang and Ghosal (2007) nonparametric Bayesian inference for Markov processes is studied, of which CPPs form a particular class, but these papers deal with estimation of the transition density of a discretely observed Markov process, which is different from the problem we consider here. A parametric Bayesian approach to inference for CPPs is studied in Insua et al. (2012, Sects. 5.5 and 10.3).
The main result of our paper is Theorem 1, in which we state sufficient conditions on the prior that yield a posterior rate of contraction of the order \((\log ^{\kappa }(n\Delta ))/ \sqrt{n\Delta },\) for some constant \(\kappa >0.\) We argue that this rate is a nearly (up to a logarithmic factor) optimal posterior contraction rate in our problem. Our main result complements the one in Gugushvili et al. (2015), in that it treats both the low and high frequency observation schemes simultaneously, with emphasis on the latter. We note (again) a fundamental difference between the present paper and Gugushvili et al. (2015), when it comes down to the techniques to prove the main result. As Theorem 2.1 of Ghosal et al. (2000) cannot immediately be used, we take an alternative tour that avoids this theorem, but instead refines a number of technical results involving properties of statistical tests that form essential ingredients of the proof in Ghosal et al. (2000). These refined results are then used as key technical steps in a direct proof of our Theorem 1. Furthermore, it establishes the posterior contraction rate for infinitely smooth jump size densities \(f_0,\) which is not covered by Gugushvili et al. (2015). On the other hand, Gugushvili et al. (2015) deals with multidimensional CPPs, while in this paper we consider only the onedimensional case. Finally, in this work we also discuss a practical implementation of our nonparametric Bayesian approach. The computational problem is dealt with by inclusion of auxiliary variables. More precisely, we show how a Markov chain Monte Carlo algorithm can be devised that samples from the joint distribution of the unknown parameters in the mixture density and the introduced auxiliary variables. Numerical examples illustrate the feasibility of this approach.
Organisation
The remainder of the paper is organised as follows. In the next section we state some preliminaries on the likelihood, prior and notation. In Sect. 3 we first motivate the use of the scaled Hellinger metric to define neighbourhoods for which posterior contraction rate is derived in case the observations are sampled at high frequency. Then we present the main result on the posterior contraction rate (Theorem 1), whose proof is given in Sect. 5. We discuss the numerical implementation of our results in Sect. 4. Technical lemmas and their proofs used to prove the main theorem are gathered in the Appendix.
Preliminaries and notation
Likelihood, prior and posterior
We are interested in Bayesian inference with Bayes’ formula. Therefore we need to specify the likelihood in our model. We use the following notation:
The characteristic function of the Poisson sum \(Z^{\Delta }\) defined in (2) is given by
where \(\phi _{f}\) is the characteristic function of f. This can be rewritten as
which, using the fact that \(\phi _{f}\) vanishes at infinity, shows that the distribution of \(Z^{\Delta }\) is a mixture of a point mass at zero and an absolutely continuous distribution. Letting \(t\rightarrow \infty ,\) we get that \(\phi (t)\rightarrow e^{\lambda \Delta }.\) Hence \(\lambda \) is identifiable from the law of \(Z^{\Delta },\) and then so is f. The density of the law \(\mathbb {Q}_{\lambda , f}^{\Delta }\) of \(Z^{\Delta }\) with respect to the measure \(\mu ,\) which is the sum of Lebesgue measure and the Dirac measure concentrated at zero, can in fact be written explicitly as (cf. van Es et al. 2007, p. 681 and Proposition 2.1 in Duval 2013)
where \(\mathbf {1}_A\) denotes the indicator of a set A,
and \(f^{*m}\) denotes the mfold convolution of f with itself. However, the expression (4) is useless for Bayesian computations. To work around this problem, we will employ a different dominating measure. Consider the law \(\mathbb {R}_{\lambda ,f}^{\Delta }\) of \((X_t,\, t\in [0,\,\Delta ]).\) By the Theorem in Skorohod (1964, p. 261) \(\mathbb {R}_{\lambda ,f}^{\Delta }\) is absolutely continuous with respect to \(\mathbb {R}_{\widetilde{\lambda },\widetilde{f}}^{\Delta }\) if and only if \(\mathbb {P}_{f}\) is absolutely continuous with respect to \(\mathbb {P}_{\widetilde{f}}\) (we of course assume that \(\lambda ,\,\widetilde{\lambda }>0\)). A simple condition to ensure the latter is to assume that \(\widetilde{f}\) is continuous and does not take the value zero on \(\mathbb {R}.\)
Define the random measure \(\mu \) by
Under \(\mathbb {R}_{{\lambda },{f}},\) the random measure \(\mu \) is a Poisson point process on \([0,\,\Delta ]\times (\mathbb {R}\setminus \{0\})\) with intensity measure \( {\Lambda }(dt,\,dx)={\lambda } dt {f}(x)dx,\) which follows, e.g., from Theorem 1 on p. 69 and Corollary on p. 64 in Skorohod (1964). By formula (46.1) on p. 262 in Skorohod (1964), we have
By Theorem 2 on p. 245 in Skorohod (1964) and Corollary 2 on p. 246 there, the density \(k_{\lambda ,f}^{\Delta }\) of \(\mathbb {Q}_{\lambda ,f}^{\Delta }\) with respect to \(\mathbb {Q}_{\widetilde{\lambda },\widetilde{f}}^{\Delta }\) is given by the conditional expectation
where the subscript in the conditional expectation operator signifies the fact that it is evaluated under the probability \(\mathbb {R}_{ \widetilde{\lambda },\widetilde{f} }^{\Delta }.\) Hence the likelihood [in the parameter pair \((\lambda ,\,f)\)] associated with the sample \(\mathcal {Z}_n^{\Delta }\) is given by the product
An advantage of specifying the likelihood in this manner is that it allows one to reduce some of the difficult computations for the laws \(\mathbb {Q}_{\lambda ,f}^{\Delta }\) to those for the laws \(\mathbb {R}_{\lambda ,f}^{\Delta },\) which are simpler.
Observe that the priors on \(\lambda \) and f indirectly induce the prior \(\Pi = \Pi _1\times \Pi _2\) on the collection of densities \(k_{\lambda ,f}^{\Delta }.\) We will indiscriminately use the symbol \(\Pi \) to signify both the prior on \((\lambda ,\,f),\) but also on the density \(k_{\lambda ,f}^{\Delta }.\) The posterior in the first case will be understood as the posterior for the pair \((\lambda ,\,f),\) while in the second case as the posterior for the density \(k_{\lambda ,f}^{\Delta }.\) We will often use the same symbol \(\Pi \) to denote the posterior distribution of \((\lambda ,\,f)\) and on the density \(k_{\lambda ,f}^{\Delta }.\) This simplifies notationally some of the formulations below.
By Bayes’ theorem, the posterior measure of any measurable set \(A\subset (0,\,\infty )\times \mathcal {F}\) is given by
Upon setting \(\overline{A}=\{ k_{\lambda ,f}{\text {:}}\,(k,\,\lambda )\in A \}\) and recalling our conventions above, this can also be written as
Once the posterior is available, one can next proceed with computation of other quantities of interest in Bayesian statistics, such as Bayes point estimates or credible sets.
Notation
Throughout the paper we will use the following notation to compare two sequences \(\{a_n\}\) and \(\{b_n\}\) of positive real numbers: \(a_n\lesssim b_n\) will mean that there exists a constant \(C>0\) that is independent of n and is such that \(a_n\le C b_n,\) while \(a_n\gtrsim b_n\) will signify the fact that \(a_n\ge C b_n.\)
Next we introduce various notions of distances between probability measures. The Hellinger distance \(h(\mathbb {Q}_{0},\,\mathbb {Q}_{1})\) between two probability laws \(\mathbb {Q}_{0}\) and \(\mathbb {Q}_{1}\) on a measurable space \((\Omega ,\,\mathfrak {F})\) is defined as
Assume further \(\mathbb {Q}_0\ll \mathbb {Q}_1.\) The Kullback–Leibler (or informational) divergence \(\mathrm {K}(\mathbb {Q}_{0},\,\mathbb {Q}_{1})\) is defined as
while the \(\mathrm {V}\)discrepancy is defined through
Here is some additional notation. For \(f,\,g\) nonnegative integrable functions, not necessarily densities, we write
Note that these ‘distances’ are all nonnegative and only zero if \(f=g\) a.e. If f and g are densities of probability measures \(\mathbb {Q}_0\) and \(\mathbb {Q}_1\) on \((\mathbb {R},\,\mathcal {B}),\) respectively, then the above ‘distances’ reduce to the previously introduced ones.
We will also use \(\mathrm {K}(x,\,y)=x\log \frac{x}{y}x+y\) for \(x,\,y>0.\) Note that also \(\mathrm {K}(x,\,y)\ge 0\) and \(\mathrm {K}(x,\,y)=0\) if and only if \(x=y.\)
Main result on posterior contraction rate
Denote the true parameter values for the CPP by \((\lambda _0,\, f_0).\) Recall that the problem is to estimate \(f_0\) and \(\lambda _0\) based on the observations \(\mathcal {Z}^\Delta _n\) and that \(\Delta \rightarrow 0\) in a high frequency regime. To say that a pair \((f,\,\lambda )\) lies in a neighbourhood of \((f_0,\,\lambda _0),\) one needs a notion of distance on the corresponding measures \(\mathbb {Q}^{\Delta }_{\lambda ,f}\) and \(\mathbb {Q}^{\Delta }_{\lambda _0,f_0},\) the two possible induced laws of \(Z^\Delta _i=X_{i\Delta }X_{(i1)\Delta }.\) The Hellinger distance is a popular and rather reasonable choice to that end in nonparametric Bayesian statistics. However, for \(\Delta \rightarrow 0\) the Hellinger metric h between those laws automatically tends to 0. The first assertion of Lemma 1 below states that \(h(\mathbb {Q}^{\Delta }_{\lambda ,f},\,\mathbb {Q}^{\Delta }_{\lambda _0,f_0})\) is of order \(\sqrt{\Delta }\) when \(\Delta \rightarrow 0.\) This motivates to replace the ordinary Hellinger metric h with the scaled metric \(h^\Delta =h/\sqrt{\Delta }\) in our asymptotic analysis for high frequency data. Of course, for fixed \(\Delta \) (in which case one can take \(\Delta =1\) w.l.o.g.), nothing changes with this replacement. The lemma also shows that the Kullback–Leibler divergence and the Vdiscrepancy are of order \(\Delta \) for \(\Delta \rightarrow 0.\) Therefore we will also use the scaled distances \(\mathrm {K}^\Delta =\mathrm {K}/\Delta \) and \(\mathrm {V}^\Delta =\mathrm {V}/\Delta \)
Lemma 1
The following expressions hold true:
The proof will be presented in the appendix.
Remark 1
The Hellinger process (here deterministic) of order \(\frac{1}{2}\) for continuous observations of X on an interval \([0,\,t]\) is given by Jacod and Shiryaev (2003, Sects. IV.3 and IV.4a)
from which it follows that \(h^2(\mathbb {R}^t_{\lambda ,f},\,\mathbb {R}^t_{\lambda _0,f_0})=22\exp (h_t),\) whose derivative in \(t=0\) is the same as in (9) and thus equal to \(2h_1.\) For the Kullback–Leibler divergence and the discrepancy \(\mathrm {V}\) similar assertions hold. These observations have the following heuristic explanation. For \(\Delta \rightarrow 0,\) there is no big difference between observing the path of X over the interval \([0,\,\Delta ]\) and \(X_\Delta ,\) as the probability of \(\{N_\Delta \ge 2\}\) is small (of order \(\Delta ^2\)).
In order to determine the posterior contraction rate in our problem, we now specify suitable neighbourhoods \(A_n\) of \((\lambda _0,\,f_0),\) for which this will be done. Let \(M>0\) be a constant and let \(\{\varepsilon _n\}\) be a sequence of positive numbers, such that \(\varepsilon _n\rightarrow 0\) as \(n\rightarrow \infty .\) Let
be a rescaled Hellinger distance. Lemma 1 suggests that this is the right scaling to use. Introduce the complements of the Hellingertype neighbourhoods of \((\lambda _0,\,f_0),\)
We shall say that \(\varepsilon _n\) is a posterior contraction rate, if there exists a constant \(M>0,\) such that
in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability as \(n\rightarrow \infty .\) Our goal in this section is to determine the ‘fastest’ rate at which \(\varepsilon _n\) is allowed to tend to zero, while not violating (12).
We will assume that the observations are generated from a CPP that satisfies the following assumption.
Assumption 1

(i)
\(\lambda _0\) is in a compact set \([\underline{\lambda },\, \overline{\lambda }]\subset (0,\,\infty );\)

(ii)
The true density \(f_0\) is a location mixture of normal densities, i.e.,
$$\begin{aligned} f_0(x)=f_{ H_0 , \sigma _0 } (x)=\int \phi _{\sigma _0 }(xz)\mathrm {d}H_0(z), \end{aligned}$$for some fixed distribution \(H_0\) and a constant \(\sigma _0 \in [\underline{\sigma },\, \overline{\sigma }]\subset (0,\,\infty ).\) Furthermore, for some \(0<\kappa _0<\infty ,\, H_0[{}\kappa _0,\,\kappa _0]=1,\) i.e., \(H_0\) has compact support.
The more general locationscale mixtures of normal densities,
possess even better approximation properties than the location mixtures of the normals (here \(H_0\) and \(K_0\) are distributions) and could also be considered in our setup. However, this would lead to additional technical complications, which could obscure essential contributions of our work.
For obtaining posterior contraction rates we need to make some assumptions on the prior.
Assumption 2

(i)
The prior on \(\lambda ,\, \Pi _1,\) has a density \(\pi _1\) (with respect to the Lebesgue measure) that is supported on the finite interval \([\underline{\lambda },\,\overline{\lambda }]\subset (0,\,\infty )\) and is such that
$$\begin{aligned} 0<\underline{\pi }_1 \le \pi _1(\lambda )\le \overline{\pi }_1<\infty , \quad \lambda \in [\underline{\lambda },\,\overline{\lambda }], \end{aligned}$$(13)for some constants \(\underline{\pi }_1\) and \(\overline{\pi }_1;\)

(ii)
The base measure \(\alpha \) of the Dirichlet process prior \(D_{\alpha }\) has a continuous density on an interval \([{}\kappa _0\zeta ,\,\kappa _0+\zeta ],\) with \(\kappa _0\) as in Assumption 1(ii), for some \(\zeta >0,\) is bounded away from zero there, and for all \(t>0\) satisfies the tail condition
$$\begin{aligned} \alpha (z>t)\lesssim e^{bt^{\delta }}, \end{aligned}$$(14)with some constants \(b>0\) and \(\delta >0;\)

(iii)
The prior on \(\sigma ,\,\Pi _3,\) is supported on the interval \([\underline{\sigma },\,\overline{\sigma }]\subset (0,\,\infty )\) and is such that its density \(\pi _3\) with respect to the Lebesgue measure satisfies
$$\begin{aligned} 0<\underline{\pi }_3 \le \pi _3(\sigma )\le \overline{\pi }_3<\infty , \quad \sigma \in [\underline{\sigma },\,\overline{\sigma }], \end{aligned}$$for some constants \(\underline{\pi }_3\) and \(\overline{\pi }_3.\)
Assumptions 1 and 2 parallel those given in Ghosal and Vaart (2001) in the context of nonparametric Bayesian density estimation using the Dirichlet location mixture of normal densities as a prior. We refer to that paper for an additional discussion.
The following is our main result. Note that it covers both the case of high frequency observations (\(\Delta \rightarrow 0\)) and observations with fixed intersampling intervals. We use \(\Pi \) to denote the posterior on \((\lambda ,\,f).\)
Theorem 1
Under Assumptions 1 and 2, provided \(n\Delta \rightarrow \infty ,\) there exists a constant \(M>0,\) such that for
we have
in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability as \(n\rightarrow \infty .\)
For fixed \(\Delta \) (w.l.o.g. one may then assume \(\Delta =1\)) the posterior contraction rate in Theorem 1 reduces to \(\varepsilon _n=\frac{\log ^{\kappa }(n)}{\sqrt{n}}.\) We also see that the posterior contraction rate is controlled by the parameter \(\delta \) of the tail behaviour in (14). Note that if (14) is satisfied for some \(\delta >4,\) it is also automatically satisfied for all \(0<\delta \le 4.\) The stronger the decay rate in (14), the better the contraction rate, but all \(\delta \ge 4\) give the same value \(\kappa =1.\) The best possible posterior contraction rate in Theorem 1 for minimal \(\delta \) is obtained for \(\delta =4.\) In the proof in Sect. 5 we can therefore assume that \(\delta \le 4.\)
As on p. 1239 in Ghosal and Vaart (2001) and similar Corollary 5.1 there, Theorem 1 implies existence of a point estimate of \((\lambda _0,\,f_0)\) with a frequentist convergence rate \(\varepsilon _n.\) The (frequentist) minimax convergence rate for estimation of \(k_{\lambda ,f}^{\Delta }\) relative to the Hellinger distance is unknown in our problem, but an analogy to Ibragimov and Khas’minskiĭ (1982) suggests that up to a logarithmic factor it should be of order \(\sqrt{n\Delta }\) (cf. Ghosal and Vaart 2001, p. 1236). The logarithmic factor is insignificant for all practical purposes. The convergence rate of an estimator of the Lévy density with loss measured in the \(L_2\)metric in a more general Lévy model than the CPP model is \((n\Delta )^{\beta /(2\beta +1)},\) whenever the target density is Sobolev smooth of order \(\beta \) (cf. Comte and GenonCatalot 2011). Our contraction rate is hence, roughly speaking, a limiting case of the convergence in Comte and GenonCatalot (2011) for \(\beta \rightarrow \infty .\)
Algorithms for drawing from the posterior
In this section we discuss computational methods for drawing from the distribution of the pair \((\lambda ,\, f),\) conditional on \(\mathcal {X}_n^\Delta \) (or equivalently: conditional on \(\mathcal {Z}_n^\Delta \)). In the following there is no specific need that the observational times are equidistant. We will assume observations at times \(0<t_1<\cdots <t_n\) and set \(\Delta _i =t_it_{i1}\, (1\le i \le n\)). Further, for consistency with notation following shortly, we set \(z_i=X_{t_i}X_{t_{i1}}\) and \(z=(z_1,\ldots , z_n).\) We will use “Bayesian notation” throughout and write p for a probability density of mass function and use \(\pi \) similarly for a prior density or mass function.
In general, it is infeasible to generate independent realisations of the posterior distribution of \((\lambda ,\,f).\) To see this: from (4) one obtains that the conditional density of a nonzero increment z on a time interval of length \(\Delta \) is given by
which generally is rather intractable due to the infinite weighted sum of convolutions. We specialise to the case where the jump size distribution is a mixture of \(J\ge 1\) Gaussians. The richness and versatility of the class of finite normal mixtures is convincingly demonstrated in Marron and Wand (1992).
Hence, we assume
where \(\phi (\cdot ;\, \mu ,\, \sigma ^2)\) denotes the density of a random variable with \(\mathcal {N}(\mu ,\, \sigma ^2)\) distribution. Note that in (16) we parametrise the density with the precision \(\tau .\) In the “simple” case \(J=2\) the convolution density of k independent jumps is given by
Plugging this expression into Eq. (15) confirms the intractable form of \(p(z\mid \lambda ,\, f).\)
We will introduce auxiliary variables to circumvent the intractable form of the likelihood. In case the CPP is observed continuously, the problem is much easier as now the continuous time likelihood on an interval \([0,\,T]\) is known to be (Shreve 2008, Theorem 11.6.7)
where the \(T_i\) are the jump times of the CPP, \(J_i\) the corresponding jump sizes and \(V=\{i{\text {:}}\, T_i \le T\}.\) The tractability of the continuous time likelihood naturally suggests the construction of a data augmentation scheme. Denote the values of the CPP in between times \(t_{i1}\) and \(t_i\) by \(x_{(i1,i)}.\) We will refer to \(x_{(i1,i)}\) as the missing values on the ith segment. Set
A data augmentation scheme now consists of augmenting auxiliary variables \(x^{mis}\) to \((\lambda ,\,f)\) and constructing a Markov chain that has \(p(x^{mis},\, \lambda ,\, f \mid z)\) as invariant distribution. More specifically, a standard implementation of this algorithm consists of the following steps:

(1)
Initialise \(x^{mis}.\)

(2)
Draw \((\lambda ,\, f) \mid (x^{mis},\, z).\)

(3)
Draw \(x^{mis} \mid (\lambda ,\, f,\, z).\)

(4)
Repeat steps 2 and 3 many times.
Under weak conditions, the iterates for \((\lambda ,\,f)\) are (dependent) draws from the posterior distribution. Step 3 entails generating compound Poisson bridges. By the Markov property, bridges on different segments can be drawn independently. Data augmentation has been used in many Bayesian computational problems, see, e.g., Tanner and Wong (1987). The outlined scheme can be applied to the problem at hand, but we explain shortly that imputation of complete CPPbridges (which is nontrivial) is unnecessary and we can do with less imputation, thereby effectively reducing the state space of the Markov chain.
As we assume that the jumps are drawn from a nonatomic distribution, imputation is only necessary on segments with nonzero increments. For this reason we let
denote the set of observations with nonzero jump sizes and define the number of segments with nonzero jumps to be \(I =\mathcal {I}.\)
Auxiliary variables
Note that if \(Y \sim f\) with f as in (16), then Y can be simulated by first drawing its label L, which equals j with probability \(\rho _j,\) and next drawing from the \(N(\mu _{L},\, 1/\tau )\) distribution. Knowing the labels, sampling the jumps conditional on their sum being z is much easier compared to the case with unknown labels. Adding auxiliary variables as labels is a standard trick used for inference in mixture models (see, e.g., Diebolt and Robert 1994; Richardsen and Green 1997). For the problem at hand, we can do with even less imputation: all we need to know is the number of jumps of each type on every segment with nonzero jump size. For \(i \in \mathcal {I}\) and \(j \in \{1,\ldots , J\},\) let \(n_{ij}\) denote the number of jumps of type j on segment i. Denote the set of all auxiliary variables by \(\mathbf {a}=\{a_i, \, i \in I\},\) where
In the following we will use the following additional notation: for \(i=1,\ldots ,n,\,j=1,\ldots , J\) we set
These are the number of jumps on the ith segment, the total number of jumps of type j (summed over all segments) and the total number of jumps of all types, respectively.
Reparametrisation and prior specification
Instead of parametrising with \((\lambda ,\, \rho _1,\ldots , \rho _J),\) we define
Then
The background of this reparametrisation is the observation that a compound Poisson random variable Z whose jumps are of J types can be decomposed as \(Z=\sum _{j=1}^JZ_j,\) where the \(Z_j\) are independent, compound Poisson random variables whose jumps are of type j only, and where the parameter of the Poisson random variable is \(\psi _j.\) In what follows we use \(\theta =(\psi ,\,\mu ,\, \tau )\) with \(\psi =(\psi _1,\ldots , \psi _J)\) and \(\mu =(\mu _1, \ldots , \mu _J).\)
Denote the Gamma distribution with shape parameter \(\alpha \) and rate \(\beta \) by \(\mathcal {G}(\alpha ,\,\beta ).\) We take priors
with positive hyperparameters \((\alpha _0,\, \beta _0,\, \alpha _1,\, \beta _1,\, \kappa )\) fixed.
Hierarchical model and data augmentation scheme
We construct a Metropolis–Hastings algorithm to draw from
For an index \(i\in \mathcal {I}\) we set \(\mathbf {a}_{i} = \{ a_j,\, j \in \mathcal {I}\setminus \{i\}\}.\) The two main steps of the algorithm are:

(i)
Update segments for each segment \(i \in \mathcal {I},\) draw \(a_i\) conditional on \((\theta ,\, z,\, \mathbf {a}_{i});\)

(ii)
Update parameters draw \(\theta \) conditional on \((z,\, \mathbf {a}).\)
Compared to the full data augmentation scheme discussed previously, the present approach is computationally much cheaper as the amount of imputation scales with the number of segments that need imputation. If the time in between observations is fixed and equal to \(\Delta ,\) then the expected number of segments for imputation equals \(n (1 e^{\lambda \Delta }),\) which is for small \(\Delta \) approximately proportional to \(n\Delta \lambda .\)
Denote the Poisson distribution with mean \(\lambda \) by \(\mathcal {P}(\lambda ).\) Including the auxiliary variables, we can write the observation model as a hierarchical model
(with \(i\in \{1,\ldots , n\}\) and \(j\in \{1,\ldots , J\}\)). This implies
Updating segments
Updating the ith segment requires drawing from
We do this with a Metropolis–Hastings step. First we draw a proposal \(n_i^\circ \) (for \(n_i\)) from a \(\mathcal {P}(\lambda \Delta _i\)) distribution, conditioned to have nonzero outcome. Next, we draw
where \(\mathcal {MN}\) denotes the multinomial distribution. Hence the proposal density equals
The acceptance probability for the proposal \(n^\circ \) equals \(1\wedge A,\) with
Updating parameters
The proof of the following lemma is given in Appendix 1.
Lemma 2
Conditional on \(\mathbf {a},\, \psi _1,\ldots , \psi _J\) are independent and
Furthermore,
where P is the symmetric \(J\times J\) matrix with elements
q is the Jdimensional vector with
\(R>0\) is given by
and \(Rq^{\prime } P^{1} q >0.\)
Remark 2
If for some \(j\in \{1,\ldots , J\}\) we have \(s_j=0\) (no jumps of type j), then the matrix \(\tilde{P}\) is singular. However, adding \(\kappa I_{J\times J}\) ensures invertibility of P.
Numerical illustrations
The first two examples concern mixtures of two normal distributions We simulated \(n=5.000\) segments with \(\Delta =1,\, \mu _1=2,\, \mu _2={}1\) and \(\tau =1.\) For the priorhyperparameters we took \(\alpha _0=\beta _0=\alpha _1=\beta _1=1,\, \xi _1=\xi _2=0\) and \(\kappa =1.\)
The results for \(\lambda \Delta =1,\, \rho _1=0.8,\, \rho _2=0.2\) and hence \(\psi _1=0.8\) and \(\psi _2=0.2\) are shown in Fig. 1. The densities obtained from the posterior mean of the parameter estimates and the true density are shown in Fig. 2. The average acceptance probability for updating the segments was 51%.
The results for \(\lambda \Delta =3,\, \rho _1=0.8,\, \rho _2=0.2\) and hence \(\psi _1=2.4\) and \(\psi _2=0.6\) are shown in Fig. 3. The densities obtained from the posterior mean of the parameter estimates and the true density are shown in Fig. 4. The average acceptance probability for updating the segments was 41%. Observe that the autocorrelation functions of the iterations of the \(\psi _i\) in the second case display a much slower decay.
We also assessed the performance of our method on a more complicated example where we took a mixture of four normals. Here \(\Delta =1,\, (\mu _1,\, \mu _2,\, \mu _3,\, \mu _4)=({}1,\, 0,\, 0.8,\, 2),\, (\psi _1,\,\psi _2,\, \psi _3,\, \psi _4)=(0.3,\, 0.4,\, 0.2,\, 0.1)\) (hence \(\lambda =1\)) and \(\tau ^{1}=0.09.\) The results obtained after simulating \(n=10.000\) segments are shown in Figs. 5 and 6.
Mixtures of normals need not be multimodal and can also yield skew densities. As an example, we consider the case where \((\mu _1,\,\mu _2)=(0,\,2),\,(\psi _1,\, \psi _2)=(1.5,\,0.5)\) (hence \(\lambda =2\)) and \(\tau =1.\) Data were generated and discretely sampled with \(\Delta =1\) and \(n=5.000\) segments. A plot of the posterior mean is shown in Fig. 7.
Discussion
As can be seen from the autocorrelation plots, mixing of the chain deteriorates when \(\lambda \Delta \) increases. As the focus in this article is on high frequency data, where there are on average only a few jumps in between observations, we do not go into details on improving the algorithm. We remark that a noncentred parametrisation (see for instance Papaspiliopoulos et al. 2007) may give more satisfactory results when \(\lambda \Delta \) is large. A non centred parametrisation can be obtained by changing the hierarchical model in (17). Denote by \(F^{1}_\lambda \) the inverse cumulative distribution function of the \(\mathcal {P}(\lambda )\) distribution. Let \(u_{ij}\) (\(i=1,\ldots , n\) and \(j=1,\ldots , J\)) be a sequence of independent \(U(0,\,1)\) random variables and set \(u=\{u_{ij},\, i=1,\ldots , n,\, j=1,\ldots , J\}.\) By considering the hierarchical model
(\(i\in \{1,\ldots , n\}\) and \(j\in \{1,\ldots , J\}),\, \psi \) can be updated using a Metropolis–Hastings step. In this way \(\{n_{ij}\}\) and \(\psi \) are updated simultaneously.
Another option is to integrate out \((\mu ,\,\tau )\) from \(p(\theta ,\,z,\,\mathbf {a}).\) In this model it is even possible to integrate out \(\psi \) as well. In that case only the auxiliary variables \(\mathbf {a}\) have to be updated. Yet another method to improve the efficiency of the algorithm is to use ideas from parallel tempering (Cf. Brooks et al. 2011, Chap. 11).
Proof of Theorem 1
There are a number of general results in Bayesian nonparametric statistics, such as the fundamental Theorem 2.1 in Ghosal et al. (2000) and Theorem 2.1 in Ghosal and Vaart (2001), which allow determination of the posterior contraction rates through checking certain conditions, but none of these results is easily and directly applicable in our case. The principle bottleneck is that a main assumption underlying these theorems is sampling from a fixed distribution, whereas in our high frequency setting, the distributions vary with \(\Delta .\) Therefore, for the clarity of exposition in the proof of our main theorem we will choose an alternative path, which consists in mimicking the main steps of the proof of Theorem 2.1, involving judiciously chosen statistical tests, as in Ghosal et al. (2000), while also employing some results on the Dirichlet location mixtures of normal densities from Ghosal and Vaart (2001). However, a significant part of technicalities we will encounter are characteristic of the decompounding problem only.
Throughout this section we assume that Assumptions 1 and 2 hold. Furthermore, in view of the discussion that followed Theorem 1 we will without loss of generality assume that \(0<\delta \le 4.\) All the technical lemmas used in this section are collected in the appendices.
We start with the decomposition
where \(0\le \phi _n\le 1\) is a sequence of tests based on observations \(\mathcal {Z}_n^{\Delta }\) and with properties to be specified below. The idea is to show that the terms on the righthand side of the above display separately converge to zero in probability. The tests \(\phi _n\) allow one to control the behaviour of the likelihood ratio
on the set where it is not wellbehaved due to the fact that \((\lambda ,\,f)\) is ‘far away’ from \((\lambda _0,\,f_0).\)
Construction of tests
The next lemma is an adaptation of Theorem 7.1 from Ghosal et al. (2000) to decompounding. A proof is given in the appendix. We use the notation \(D(\varepsilon ,\,A,\,d)\) to denote the \(\varepsilon \)packing number of a set A in a metric space with metric d, applied in our case with d the scaled Hellinger metric \(h^\Delta .\)
Lemma 3
Let \(\mathcal {Q}\) be an arbitrary set of probability measures \(\mathbb {Q}^\Delta _{\lambda ,f}.\) Suppose for some nonincreasing function \(D(\varepsilon ),\) some sequence \(\{\varepsilon _n\}\) of positive numbers and every \(\varepsilon >\varepsilon _n,\)
Then for every \(\varepsilon >\varepsilon _n\) there exists a sequence of tests \(\{\phi _n\}\) (depending on \(\varepsilon >0),\) such that
where \(K>0\) is a universal constant.
In the proofs of Propositions 1 and 2 we need the inequalities below. There exists a constant \(\overline{C}\in (0,\,\infty )\) depending on \(\underline{\lambda }\) and \(\overline{\lambda }\) only, such that for all \(\lambda _1,\,\lambda _2\in [\underline{\lambda },\,\overline{\lambda }]\) and \(f_1,\,f_2\) it holds that
These inequalities can be proven in the same way as Lemma 1 in Gugushvili et al. (2015).
Let \(\varepsilon _n\) be as in Theorem 1. Throughout, \(\overline{C}\) denotes the above constant. For a constant \(L>0\) define the sequences \(\{a_n\}\) and \(\{\eta _n\}\) by
We will show that inequality (24) holds true for every \(\varepsilon =M\varepsilon _n\) with \(M>2\) and the set of measures \(\mathcal {Q}\) equal to
As a first step, note that we have
where \(N\left( \frac{\varepsilon _n\sqrt{\Delta }}{2},\,\mathcal {Q}_n,\,h\right) \) is the covering number of the set \(\mathcal {Q}_n\) with hballs of size \(\varepsilon _n\sqrt{\Delta }/2.\) The first inequality in (28) follows from assuming \(M>2.\) For bounding the righthand side in (28), we have the following proposition.
Proposition 1
We have
Proof
Define
Let \(\{\lambda _i\}\) be centres of the balls from a minimal covering of \([\underline{\lambda },\, \overline{\lambda }]\) with \(\cdot \)balls of size \(\eta _n.\) Let \(\{f_j\}\) be centres of the balls from a minimal covering of \(\mathcal {F}_n\) with hballs of size \(\eta _n.\) For any \(\mathbb { Q}_{\lambda , f_{H,\sigma }} \in \mathcal {Q }_n,\) by (27) we have
by appropriate choices of i and j. It follows that
Evidently,
As we assume \(\delta \le 4,\) we can apply the arguments in Ghosal and van der Vaart (2001, pp. 1251–1252) see in particular formulae (5.8)–(5.10) (cf. also Theorem 3.1 and Lemma A.3 there), which yield
Combination of the above three inequalities implies the statement of the proposition.
An application of Proposition 1 to (28) gives
for some positive constant \(c_1.\) Here, the final inequality follows from our choice for \(\varepsilon _n.\) Hence, (24) is satisfied for
By Lemma 3 there exist tests \(\phi _n\) such that for all n large enough
Bound on \(\mathrm {I}_n\) in (23)
First note that by Eq. (30)
Chebyshev’s inequality implies that \(\mathrm {I}_n\) converges to zero in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability as \(n\rightarrow \infty ,\) as soon as M is chosen so large that \(KM^2c_1>0.\,\square \)
Bound on \(\mathrm {II}_n\)
Now we consider \(\mathrm {II}_n.\) We have
We will show that the numerator \(\mathrm {III}_n\) goes exponentially fast to zero, in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability, while the denominator \(\mathrm {IV}_n\) is bounded from below by an exponential function, with \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability tending to one, in such a way that the ratio of \(\mathrm {III}_n\) and \(\mathrm {IV}_n\) still goes to zero in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability.
Bounding \(\mathrm {III}_n\)
As \(1_{\{A(\varepsilon _n,M)\}} \le 1_{\mathcal {Q}_n^c}+1_{\{A(\varepsilon _n,M)\cap \mathcal {Q}_n\}}\) we have
Here we applied Fubini’s theorem to obtain the second term on the righthandside, which by (31) is bounded by \(\exp (KM^2n\Delta \varepsilon _n^2).\) Furthermore,
where the last inequality is formula (5.11) in Ghosal and Vaart (2001). Hence
Bounding \(\mathrm {IV}_n\)
Recall \(\mathrm {K}_\Delta =\mathrm {K}/\Delta \) and \(\mathrm {V}_\Delta =\mathrm {V}/\Delta .\) Let
and
Note that \(n\Delta \widetilde{\varepsilon }_n^2\rightarrow \infty \) when \(n\rightarrow \infty .\)
We will use the following bound, an adaptation of Lemma 8.1 in Ghosal et al. (2000) to our setting, valid for every \(\varepsilon >0\) and \(C>0,\)
where
is a normalised restriction of \(\Pi (\cdot )\) to \(B^{\Delta }(\varepsilon ,\,(\lambda _0,\,f_0)).\)
By virtue of (33), with \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability tending to one, for any constant \(C>0\) we have
We will now work out the product probability on the righthand side of this inequality.
Proposition 2
It holds that
for some constant \(\bar{c}.\)
Proof
Let \(0<c\le 1/\sqrt{5\overline{C}}\) be a constant. Here \(\overline{C}\) is the constant in (25) and (26). By these inequalities it is readily seen that
It then follows by the independence assumption on \(\Pi _1\) and \(\Pi _2\) that
For the first factor on the righthand side we have by (13) that
As far as the second factor is concerned, for some constants \(\overline{c}_1,\,\overline{c}_2\) it is bounded from below by
by the same arguments as in inequality (5.17) in Ghosal and Vaart (2001). The result now follows by combining the two lower bounds.
Combining (34) with Proposition 2, with \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability tending to one as \(n\rightarrow \infty ,\) for any constant \(C>0\) we have
We are now ready for showing the final steps of proving that \(\mathrm {II}_n\) tends to zero in \(\mathbb {Q}_{\lambda _0,f_0}^{\Delta , n}\)probability. Let \(G_n\) denote the set on which inequality (35) is true. Then by (32) we obtain
Recall that \(n\Delta \widetilde{\varepsilon }_n^2=\log ^2(n\Delta ).\) Hence, the exponent in the first factor of this display is of order \(\log ^2 (n\Delta ).\) Furthermore \(a_n^\delta =L^\delta \log ^2({4\overline{C}}/{\varepsilon _n}),\) which is of order \(\log ^2(n\Delta )\) as well. It follows that, provided the constants L and M are chosen large enough, the righthand side of the above display converges to zero as \(n\rightarrow \infty .\) Chebyshev’s inequality then implies that \(\mathrm {II}_n\) converges to zero in probability as \(n\rightarrow \infty .\) This completes the proof of Theorem 1. \(\square \)
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Acknowledgements
We wish to thank Wikash Sewlal from Delft University of Technology for the simulation results of the example with a mixture of four normals and the skewed density. The research leading to these results has received funding from the European Research Council under ERC Grant Agreement 320637.
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Additional lemmas and proofs
Additional lemmas and proofs
Proof of Lemma 1
We give a detailed proof of equality (9). As we are interested in small values of \(\Delta ,\) we make some necessary approximations. Starting point is the expansion for the ‘density’ of \(\mathbb {Q}^\Delta _{\lambda ,f}\) with respect to the Lebesgue measure,
see (4), with coefficients \(a_m\) defined in (5). It follows that we have the likelihood ratio
where we collected terms of order \(\Delta ^m\) for \(m\ge 2\) as \(o(\Delta ).\) Hence we get for the Hellinger affinity
the approximating expression
It follows that for \(\Delta \rightarrow 0,\)
Hence, for \(\Delta \rightarrow 0,\)
Equality (9) follows. The proofs of the equalities (10) and (11) follow a similar line of reasoning.
Proof of Lemma 3
The proof is an adaptation of Theorem 7.1 from Ghosal et al. (2000) to decompounding. In all what follows it is assumed that \(\mathbb {Q}^\Delta _{\lambda ,f}\in \mathcal {Q},\) but we suppress this assumption in the notation. Observe that
From this point on the arguments from the proof of Theorem 7.1 in Ghosal et al. (2000) are applicable (with \(\varepsilon \) replaced by \(\varepsilon \sqrt{\Delta }\)) and eventually lead to the desired result. The role of formulae (7.1)–(7.2) in that proof are played in the present context by (36) and (37) below.
For a given \((\lambda _1,\,f_1)\) there exists a sequence of tests \(\phi _n\) based on \(\mathcal {Z}_n^{\Delta },\) such that
These two inequalities simply follow by rewriting the inequalities
which are proved in Ghosal et al. (2000, pp. 520–521) and rely upon the results in Birgé (1984) and Cam (1986).
Proof of Lemma 2
As the priors for \(\psi _1,\ldots , \psi _J\) are independent, we obtain that
which proves the first statement of the lemma.
For \((\mu ,\, \tau )\) we get
This is proportional to
where
From this expression it is easily seen that we can integrate out \(\mu \) to obtain the distribution of \(\tau ,\) conditional on \((z,\, \mathbf {a}).\) To get this right, write \(D(\mu )\) as a quadratic form of \(\mu {\text {:}}\)
By completing the square, we find that
The integrand is (up to a proportionality constant), the density of a bivariate normal random vector with mean vector \(P^{1}q\) and covariance matrix \(\tau ^{1} P^{1}\) evaluated in \(\mu .\) This implies that the preceding display equals
We conclude that
which proves the asserted Gamma distribution of \(\tau .\) This computation also immediately leads to the assertion on the distribution of \(\mu .\) We finally show that the rate parameter appearing for \(\tau \) is positive. By definition \(D(\mu )\ge 0\) for all \(\mu .\) This implies that \(D(P^{1} q)=q^{\prime } P^{1} q2 q^{\prime } P^{1}q+R=Rq^{\prime } P^{1} q\ge 0.\)
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Gugushvili, S., van der Meulen, F. & Spreij, P. A nonparametric Bayesian approach to decompounding from high frequency data. Stat Inference Stoch Process 21, 53–79 (2018). https://doi.org/10.1007/s1120301691531
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Keywords
 Compound Poisson process
 Nonparametric Bayesian estimation
 Posterior contraction rate
 High frequency observations
Mathematics Subject Classification
 Primary: 62G20
 Secondary: 62M30