A Multivariate Claim Count Model for Applications in Insurance by Daniela Anna Selch & Matthias Scherer

Author:Daniela Anna Selch & Matthias Scherer
Language: eng
Format: epub
ISBN: 9783319928685
Publisher: Springer International Publishing

3.1.2 Continuous Monitoring

Ideally, the claim number process is monitored continuously up to a finite time horizon , providing one observation of on [0, T]. As is a Poison cluster process, the arrival times and jump sizes of each cluster of claims are recorded in this scenario. Let be the total number of incoming clusters in the time period [0, T], the successive cluster arrival times, and the corresponding cluster sizes. From these data, a discrete observation of can be extracted and the previously discussed Methods (M1) and (M2) can be applied. To exploit all available information, however, two more estimation methods are proposed in the following based on the Poisson cluster process representation of .

Firstly, maximum likelihood estimation is used jointly for the iid cluster inter-arrival times and the iid cluster sizes, cf. the approach in [8, Sect. 6.4, pp. 105] for univariate compound Poisson processes. The maximum likelihood method requires no iid samples, it is sufficient that the joint likelihood function of the observations is available, i.e. the probability of observing the given data in the model, which reduces for point observations of continuous distributions to the mass of the density function at the observation point. For a general compound Poisson process, where the jump size distribution is specified independently of the Poisson arrival process, maximum likelihood estimation can be performed separately for both parts. In the given setting, however, the parameters affect both jump size and jump intensity, making a joint optimization necessary.

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