Fixed-k Asymptotic Inference about Tail Properties

We consider inference about tail properties of a distribution from an i.i.d. sample, based on extreme value theory. All of the numerous previous suggestions rely on asymptotics where eventually, an in…nite number of observations from the tail behave as predicted by extreme value theory, enabling the consistent estimation of the key tail index, and the construction of con…dence intervals using the delta method or other classic approaches. In small samples, however, extreme value theory might well provide good approximations for only a relatively small number of tail observations. To accommodate this concern, we develop asymptotically valid con…dence intervals for high quantile and tail conditional expectations that only require extreme value theory to hold for the largest k observations, for a given and …xed k. Small sample simulations show that these “…xed-k” intervals have excellent small sample coverage properties, and we illustrate their use with mainland U.S. hurricane data. In addition, we provide an analytical result about the additional asymptotic robustness of the …xedk approach compared to any approach that relies on consistent estimation of the tail index parameter.