Transformed Beta and Gamma Distributions and Aggregate Losses [Discussion]

One of the most important problems in collective risk theory has been the computation of the distribution of aggregate losses given individual frequency and severity distributions. Various approaches have been tried since the subject was first introduced by Filip Lundberg more than seventy-five years ago (Cramer [I]). These include approximation, simulation, and actual computation using numerical techniques. (A stochastic approach is also possible and the reviewers hope to discuss this in a later paper.) Approximations have been used with mixed success over the years. An appeal to the central limit theorem "justifies" a normal approximation if the number of claims is large (Beard, Pentikainen, Pesonen 121). This has not been satisfactory. Other approximations, such as normal power, Esscher, Gamma, Pareto, and just about any other distribution, have been used based on various theoretical (we can "prove" it) or empirical (it works) arguments. The use of these approximations has not been entirely satisfactory. The reviewers offer a reason for this later.