| Truncated normal |
| Written by theoretic |
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A standard claim size distribution is the truncated normal. This means that the Xj’s have distribution function F(x) = P(\Y\ < x) for a normally distributed random variable Y. If we assume Y standard normal, F(x) = 2 (Ф(х) — 0.5) for x > 0, where Ф is the standard normal distribution function with density The latter relation is often referred to as Mill’s ratio. With Mill’s ratio in mind, it is easy to verify that the truncated normal distribution is light-tailed. Using an analogous argument, it can be shown that the gamma distribution, for any choice of parameters, is light-tailed. Verify this. A typical example of a heavy-tailed claim size distribution is the Pareto distribution with tail parameter a > 0 and scale parameter к > 0, given by ( ка b x)= — , x > 0 . (n + x)a Another prominent heavy-tailed distribution is the Weibull distribution with shape parameter т < 1 and scale parameter с > 0: F(x)= e ~cx , x > 0 . However, for т > 1 the Weibull distribution is light-tailed. We refer to Tables 3.2.17 and 3.2.19 for more distributions used in insurance practice. |