Composite goodness-of-fit tests for left-truncated loss samples

In many financial models, such as those addressing value at risk and ruin probabilities, the accuracy of the fitted loss distribution in the upper tail of the loss data is crucial. In such situations, it is important to test the fitted loss distribution for the goodness of fit in the upper quantiles, while giving lesser importance to the fit in the low quantiles and the center of the distribution of the data. Additionally, in many loss models the recorded data are left truncated with the number of missing data unknown. We address this gap in literature by proposing appropriate goodness-of-fit tests. We derive the exact formulae for several goodness-of-fit statistics that should be applied to loss models with left-truncated data where the fit of a distribution in the right tail of the dstribution is of central importance. We apply the proposed tests to real financial losses, using a variety of distributions fitted to operational loss and the natural catastrophe insurance claims data, which are subject to the recording thresholds of $1 and $25 million, respectively.