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

In many loss models the recorded data are left-truncated with an unknown number of missing data. We derive the exact formulae for several goodness-of-fit statistics that should be applied to such models. We additionally propose two new statistics to test the fit in the upper quantiles, applicable to models where the accuracy of the upper tail estimate is crucial, as in models addressing the Value-at-Risk and ruin probabilities. We apply the tests on a variety of distributions fitted to the external operational loss and the natural catastrophe insurance claim data, subject to the recording thresholds of $1 and $25 million.