Abstract
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual claim variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modeling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importance sampling using an exponential change of measure. We conclude by numerical experiments of these algorithms, based on real car insurance claim data.
Highlights
We assume that the individual claim variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function
We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution
Let Y1, Y2, . . . be i.i.d. positive random variables representing the individual claims at an insurance company, and let N ∈ N0 = {0, 1, . . .} designate the total number of claims occurring during a certain time period, where N and the Yi’s are independent
Summary
Our objective is to execute numerical computations of the insurance risk factor, for which we consider using Monte Carlo simulations This was motivated because there are no workable analytic expressions available for the probability functions of the Abel, strict arcsine arcsine and Takacs distributions. Please note that when we write positive stable distributions (for which no integer moments exist) we mean NEF’s generated by positive stable distributions (for which all moments exist) In this way, our aggregate models become Tweedie models in the sense that both the distributions of the counting number and the distributions of the claim size belong to NEF’s (Smyth and Jorgensen, 2002; Dunn and Smyth, 2005, 2008). We show how these risks for large levels can be computed efficiently by an appropriate change of measure for importance sampling. The worst fit has been obtained for pairs of the Poisson along with the gamma, inverse Gaussian and positive stable distributions with p-value less than .00001, see Table 3 of Section 6 for a reference
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