Note Concerning the Distribution Function of the Total Loss Excluding the Largest Individual Claims

The theory of extreme values is a special branch of mathematical statistics and was mainly treated by E. J. Gumbel [4]). This theory has only been applied in a few cases to problems in the insurance business. The first practical application to insurance known to the author of the present paper is due to A. Thépaut who has invented a new reinsurance system called ECOMOR [5]. According to this system the reinsurer covers the excess risk for the m largest claims and the ceding company retains an amount equal to the (m + I) largest claim. The credit for having pointed out the importance of the theory of extreme values belongs to R. E. Beard [1]. Recently E. Franckx [3] has found a most remarkable result by disclosing the general form of the distribution for the largest claim occurring in a certain accounting period.The present paper starts from the consideration that not only is the distribution of major claims, which might be eliminated by means of reinsurance, of interest to an insurer but also the distribution of the remaining total loss after excluding the largest claims. The nature of this distribution is important not only in connection with stability and security, but also for statistical investigations of the observed claim ratio. The credibility of such an investigation might be greatly improved if a suitable number of major claims were excluded. To simplify matters, the present paper considers the case where only the largest claim is excluded.