On the estimation of the variability in the distribution tail

We propose a new measure of variability in the tail of a distribution by applying a Box–Cox transformation of parameter $$p \ge 0$$ p ≥ 0 to the tail-Gini functional. It is shown that the so-called Box–Cox Tail Gini Variability measure is a valid variability measure whose condition of existence may be as weak as necessary thanks to the tuning parameter p. The tail behaviour of the measure is investigated under a general extreme-value condition on the distribution tail. We then show how to estimate the Box–Cox Tail Gini Variability measure within the range of the data. These methods provide us with basic estimators that are then extrapolated using the extreme-value assumption to estimate the variability in the very far tails. The finite sample behaviour of the estimators is illustrated both on simulated and real data.