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In Chapter 2 we learned about three of the most prominent claim number processes, N: the Poisson process in Section 2.1, the renewal process in Section 2.2, and the mixed Poisson process in Section 2.3. In this section we take a closer look at the total claim amount process, as introduced on p. 8:
N(t) s(t)=2_]Xi, t>0, (3.0.1)i=i
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Some realistic claim size |
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We continue in Section 3.2 by considering some realistic claim size distributions and their properties. We consider exploratory statistical tools (QQ-plots, mean excess function) and apply them to real-life claim size data in order to get a preliminary understanding of which distributions fit real-life data. In this context, the issue of modeling large claims deserves particular attention.
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By applying characteristic function techniques, we learn about mixture distributions as useful tools in the context of compound Poisson and compound geometric processes. We show that the summation of independent compound Poisson processes yields a compound Poisson process and we investigate consequences of this result.
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The Order of Magnitude of the Total Claim Amount |
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Given a particular model for S, one of the important questions for an insurance company is to determine the order of magnitude of S(t). This information is needed in order to determine a premium which covers the losses represented by S(t).
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Expectation of S(t) in the Cramer-Lundberg and renewal models |
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In the Cramer-Lundberg model, EN(t) = At, where A is the intensity of the
homogeneous Poisson process N. Hence
ES(t) = \t EX1 .
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Variance of S(t) in the Cramer-Lundberg and renewal models |
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Written by theoretic
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In the Cramer-Lundberg model the Poisson distribution of N(t) gives us EN(t) = var(N(t)) = At. Hence
var(£(£)) = At [var(Xi) + (EXi)2] = At E(X2).
In the renewal model we again depend on some asymptotic formulae for EN(t) and var(N(t)); see Theorem 2.2.7 and Proposition 2.2.10:
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Expectation and variance of the total claim amount in the renewal model |
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In the renewal model, if EW\ = A-1 and EX\ are finite,
ES(t) lim = \EX\, t^oo t and if var(W\) and var(Xi) are finite, var(S(t)) г , о-,
lim = A var(Ai)+var( И) А М) . t^oo t
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The Asymptotic Behavior in the Renewal Model |
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Written by theoretic
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In this section we are interested in the asymptotic behavior of the total claim amount process. Throughout we assume the renewal model (see p. 77) for the total claim amount process S.
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The strong law of large numbers and the central limit theorem in the renewal model |
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Assume the renewal model for S.
(1) If the inter-arrival times Wi and the claim sizes Xi have finite expectation, S satisfies the strong law of large numbers:
S(t) lim = AMi a.s.t^oo t
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