Fixed-k Asymptotic Inference About Tail Properties

ABSTRACT

We consider inference about tail properties of a distribution from an iid sample, based on extreme value theory. All of the numerous previous suggestions rely on asymptotics where eventually, an infinite 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 confidence 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 confidence 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 fixed k. Small-sample simulations show that these “fixed-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 fixed-k approach compared to k_n → ∞ inference.

Keywords:

• Extreme quantiles
• tail conditional expectations
• extreme value distribution

Acknowledgment

We thank two anonymous referees, Marco del Negro and participants at workshops at Brown, Princeton and New York University for useful comments and advice.

Funding

Müller gratefully acknowledges financial support by the NSF via grant SES-1627660.

Funding

Müller gratefully acknowledges financial support by the NSF via grant SES-1627660.

Notes

* Theorem 3 in Müller and Norets (2016) provides a corresponding formal result.

† The convergence formally follows from the dominated convergence theorem for ξ < 0 (since F then has bounded support), and by Karamata's Theorem for ξ > 0, as in Equation (1.4) of Zhu and Li (2012).

‡ Not doing so leads to intervals that are very much longer on average, and with typically no better (but often) worse coverage properties.