On the problem of discriminating between the tails of distributions

In areas such as financial and insurance risk and communication network design the heaviness of the tail of the underlying distribution is crucial for the calculations. However, although it seems straightforward theoretically to distinguish between (say) exponential tails and power tails, this requires unexpectedly large samples in practice. Here we will use quantiles to compare the tails of distributions which are standardised to unit interquartile range to allow for possible infinite variance. We present some chi-squared tests of goodnessof- fit focussed on the tails. We also provide methods of quick comparison of distributions using counts over high thresholds and using extreme values.