Distribution Free Approximations in Applied Risk Theory

Most large insurance companies have today electronic computers that enable not only efficient actuarial statistics, but also research in applied risk theory. An important task in this latter field is the developing of an information system for control of the business as to the statistical balance between premiums and claims. The entire system can be separated into two parts, one descriptive and one analytic part. The descriptive part, that also may be called statistics production, is the base of the whole system and must be constructed in a general way to make it possible to apply mathematical tools in risk analysis. For the analytic part and its applications for computers there is a growing interest among actuaries as can be noticed from the reports in actuarial journals. The classical models of collective risk theory have recently been extensively illustrated by numerical calculations performed by the convolution committee in Sweden.When starting to construct the analytical part of the information system one finds that in spite of programming for the computer there is firstly a hard work to find realistic mathematical models, especially mathematical expressions for claim distributions, and secondly to estimate their parameters. According to the mathematical theory of risk, or the risk process, it is necessary to assume and test specific mathematical forms of two distributions, the number of claims and claims amount.