| Abstract: | Abstract The aim of this paper is to analyse two functions that are of general interest in the collective risk theory, namely F, the distribution function of the total amount of claims, and II, the Stop Loss premium. Section 2 presents certain basic formulae. Sections 17-18 present five claim distributions. Corresponding to these claim distributions, the functions F and II were calculated under various assumptions as to the distribution of the number of claims. These calculations were performed on an electronic computer and the numerical method used for this purpose is presented in sections 9, 19 and 20 under the name of the C-method which method has the advantage of furnishing upper and lower limits of the quantities under estimation. The means of these limits, in the following regarded as the “exact” results, are given in Tables 4-20. Sections 11-16 present certain approximation methods. The N-method of section 11 is an Edgeworth expansion, while the G-method given in section 12 is an approximation by a Pearson type III curve. The methods presented in sections 13-16, and denoted AI-A4, are all applications and modifications of the Esscher method. These approximation methods have been applied for the calculation of F and II in the cases mentioned above in which “exact” results were obtained. The results are given in Tables 4-20. The object of this investigation was to obtain information as to the precision of the approximation methods in question, and to compare their relative merits. These results arc discussed in sections 21-24. |
