Loss data analysis: Analysis of the sample dependence in density reconstruction by maxentropic methods

The problem of determining probability densities of positive random variables from empirical data is important in many fields, in particular in insurance and risk analysis. The method of maximum entropy has proven to be a powerful tool to determine probability densities from a few values of its Laplace transform. This is so even when the amount of data to compute numerically the Laplace transform is small. But in this case, the variability of the reconstruction due to the sample variability in the available data can lead to quite different results. It is the purpose of this note to quantify as much as possible the variability of the densities reconstructed by means of two maxentropic methods: the standard maximum entropy method and its extension to incorporate data with errors.The issues that we consider are of special interest for the advanced measurement approach in operational risk, which is based on loss data analysis to determine regulatory capital, as well as to determine the loss distribution of risks that occur with low frequency.