Abstract
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 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{nDelta }-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 nDelta 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.
Highlights
1.1 Problem formulationLet N = (Nt, t ≥ 0) be a Poisson process with a constant intensity λ > 0 and let Y1, Y2, Y3, . . . be a sequence of independent random variables independent of N and having a common distribution function F with density f
Suppose that corresponding to the ‘true’ parameter values λ = λ0 and f = f0, a discrete time sample X, X2, . . . , Xn is available from (1), where > 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
Each Zi has the same distribution as the random variable
Summary
Xn is available from (1), where > 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. 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 is varying, with the Dirac. The theoretical contribution of the present paper is the statement of the main result itself, and its proof To this we discuss a practical implementation of our non-parametric 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
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