Maximum Entropy Methods for Loss Data Analysis: Aggregation and Disaggregation Problems

The analysis of loss data is of utmost interest in many branches of the financial and insurance industries, in structural engineering and in operation research, among others. In the financial industry, the determination of the distribution of losses is the first step to take to compute regulatory risk capitals; in insurance we need the distribution of losses to determine the risk premia. In reliability analysis one needs to determine the distribution of accumulated damage or the first time of occurrence of a composite event, and so on. Not only that, but in some cases we have data on the aggregate risk, but we happen to be interested in determining the statistical nature of the different types of events that contribute to the aggregate loss. Even though in many of these branches of activity one may have good theoretical descriptions of the underlying processes, the nature of the problems is such that we must resort to numerical methods to actually compute the loss distributions. Besides being able to determine numerically the distribution of losses, we also need to assess the dependence of the distribution of losses and that of the quantities computed with it, on the empirical data. It is the purpose of this note to illustrate the how the maximum entropy method and its extensions can be used to deal with the various issues that come up in the computation of the distribution of losses. These methods prove to be robust and allow for extensions to the case when the data has measurement errors and/or is given up to an interval.