Loss Models: From Data to Decisions, 4th Edition

3.4 Tails of distributions

The tail of a distribution (more properly, the right tail) is the portion of the distribution corresponding to large values of the random variable. Understanding large possible loss values is important because these have the greatest impact on the total of losses. Random variables that tend to assign higher probabilities to larger values are said to be heavier tailed. Tail weight can be a relative concept (model A has a heavier tail than model B) or an absolute concept (distributions with a certain property are classified as heavy tailed). When choosing models, tail weight can help narrow the choices or can confirm a choice for a model.

3.4.1 Classification based on moments

Recall that in the continuous case the kth raw moment for a random variable that takes on only positive values (like most insurance payment variables) is given by ƒ^∞₀ x^k f(x)dx. Depending on the density function and the value of k, this integral may not exist (i.e., it may be infinite). One way of classifying distributions is on the basis of whether all moments exist. It is generally agreed that the existence of all positive moments indicates a (relatively) light right tail, while the existence of only positive moments up to a certain value (or existence of no positive moments at all) indicates a heavy right tail.

EXAMPLE 3.9

Demonstrate that for the gamma distribution all positive ...