Some notes on the favorable estimation for fitting heavy tailed data

Assessment of compound sums has many applications in insurance, auditing and operation risk capital assessment among others. We study the behavior of the total claim amount with claims taken from a homogeneous portfolio. Actuaries distinguish several types of distributions to t loss data: gamma, log-gamma, log-normal, gamma + log-gamma, gamma + log-normal and Pareto being the most important. We discuss some problems that one can encounter when misemploying the log-normal assumption based methods supported by Basel II framework. The compound sums are demonstrated to be highly sensitive on the individual claims distributions and thus a robust approach is needed. New estimators based on a robusti¯ed Johnson score are introduced and compared with the classical estimators maximal likelihood and moment estimators) and with recently introduced robust estimators of "generalized median" and "trimmed mean" type. We derive the exact distribution of the likelihood ratio tests of homogeneity and tail index of the two-parameter Pareto model which support the assessment of performance of estimators. The real data example illustrates the concepts.