Decompounding: an estimation problem for Poisson random sums

Abstract

Given a sample from a compound Poisson distribution, we consider estimation of the corresponding rate parameter and base distribution. This has applications in insurance mathematics and queueing theory. We propose a plug-in type estimator that is based on a suitable inversion of the compounding operation. Asymptotic results for this estimator are obtained via a local analysis of the decompounding functional.

Approximate Bayesian Computations to fit and compare insurance loss models

“…This problem is also interesting in the field of queueing theory to draw inference on the job size distribution when only having access to the workload. Traditionally, a decompounding (as coined by Buchmann and Grübel [6]) method builds a non-parametric estimate of the claim severity distribution based on the observations of the aggregated sums, see for instance van Es et al [39], Coca [7] and Gugushvili et al [18]. The method we propose effectively decompound the random sum but assumes that the jump sizes are driven by a parametric model.…”

Approximate Bayesian Computations to fit and compare insurance loss models

“…When the summary is the aggregated loss Ψ(n, u) = n i=1 u i , we effectively decompound the random sum. Traditionally, a decompounding method builds a non-parametric estimate of the claim severity distribution based on the observations of the aggregated sums, see Buchmann and Grübel [7] or Bøgsted and Pitts [6]. A popular application is the study of discretely observed compound Poisson processes, see for instance van Es et al [32], Coca [8] and Gugushvili et al [17] where a Bayesian non-parametric approach is used.…”

Decompounding on Compact Lie Groups

“…random variables. Estimating the distribution of the x n is known as decompounding and has been well-studied [1,2]. In the present paper, decompounding techniques are generalised to the case when G is a noncommutative group.…”

“…The classical problem of decompounding arises in the context of these processes. A functional approach to this problem is given by Buchman and Grübel [1]. A characteristic function method is studied by Van Es et al [2].…”

“…A characteristic function method is studied by Van Es et al [2]. The applications of decompounding in queuing problems and risk theory are referenced in [1]. We generalize this problem by considering decompounding on compact Lie groups.…”