| Forgetfulness property |
| Written by theoretic |
|
This property is another manifestation of the forgetfulness property of the exponential distribution; see p. 26. Indeed, the tail of the excess distribution function of Y satisfies P(Y > u + x\Y > u) = P(Y > x), x > 0. This means that this distribution function corresponds to an Exp(A) random variable; it does not depend on the threshold и Property (3.2.12) makes the exponential distribution unique: it offers another way of discriminating between heavy- and light-tailed distributions of random variables which are unbounded to the right. Indeed, if eF(u) converged to infinity for и —> oo, we could call F heavy-tailed, if eF(u) converged to a finite constant as и —> oo, we could call F light-tailed. In an insurance context this is quite a sensible definition since unlimited growth of eF(u) expresses the danger of the underlying distribution F in its right tail, where the large claims come from: given the claim size Xi exceeded the high threshold u, it is very likely that future claim sizes pierce an even higher threshold. On the other hand, for a light-tailed distribution F, the expectation of the excess (Xi — u)+ (here x+ = max(0, ж)) converges to zero (as for the truncated normal distribution) or to a positive constant (as in the exponential case), given Xi > и and the threshold и increases to infinity. This means that claim sizes with light-tailed distributions are much less dangerous (costly) than heavy-tailed distributions. In Table 3.2.9 we give the mean excess functions of some standard claim size distributions. In Figure 3.2.8 we illustrate the qualitative behavior of eF(u) for large u. |