Skip to Main Content
The online home for the publications of the American Statistical Association
482
Views
10
CrossRef citations to date
0
Altmetric
Theory and Methods

Fixed-k Asymptotic Inference About Tail Properties

&
Pages 1334-1343 | Received 01 Feb 2016, Accepted 01 Jul 2016, Published online: 13 Jun 2017
 

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 kn → ∞ inference.

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 (Citation2016) 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 (Citation2012).

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

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.