6.6 Truncation and modification at zero
At times, the distributions discussed previously do not adequately describe the characteristics of some data sets encountered in practice. This may be because the tail of the negative binomial is not heavy enough or because the distributions in the (a, b, 0) class cannot capture the shape of the data set in some other part of the distribution.
In this section, we address the problem of a poor fit at the left-hand end of the distribution, in particular the probability at zero.
For insurance count data, the probability at zero is the probability that no claims occur during the period under study. For applications in insurance where the probability of occurrence of a loss is very small, the probability at zero has the largest value. Thus, it is important to pay special attention to the fit at this point.
There are also situations that naturally occur which generate unusually large probabilities at zero. Consider the case of group dental insurance. If, in a family, both husband and wife have coverage with their employer-sponsored plans and both group insurance contracts provide coverage for all family members, the claims will be made to the insurer of the plan that provides the better benefits, and no claims may be made under the other contract. Then, in conducting studies for a specific insurer, one may find a higher than expected number of individuals who made no claim.
Similarly, it is possible to have situations in which there is less than ...