Transformed Beta and Gamma Distributions and Aggregate Losses

Distribution functions are introduced based on power transformations of beta and gamma distributions, and properties of these distributions are discussed. The gamma, beta, F. Pareto, Burr, Weibull and loglogistic distributions are special cases. The transformed gamma mixed with a gamma yields a transformed beta.The transformed gamma is used to model aggregate distributions by matching moments. The transformed beta is used to account for parameter uncertainty in this model. Calculation procedures are discussed and APL program listings are included.The transformed gamma is compared to exact methods of computing the aggregate distribution function based on the entire frequency and severity distributions.